coherence-theory-of-everything-817a4df8-293a-4f31-a32e-bd34c1f2d27b

Coherence Theory

Deriving all of modern physics from a single meta-evolutionary process

By Vladimir Ilinov

Introduction

Everything in reality emerges from a single meta-evolutionary process: Coherence. Assume this process and you can derive and unify spacetime (GR), quantum mechanics, and the Standard Model from first principles.

The Law of Coherence

A^{⋆} \in \arg max_{A} \underset{keeps working when poked}{\underset{⏟}{inf_{p \in P} H o l d (Π_{A}, K_{A, p} Π_{A})}} - λ_{f u e l} B_{f u e l} (A) - λ_{c p x} B_{c p x} (A) - λ_{l e a k} B_{l e a k} (A)

Pick the pattern that still works after every shove, charging it for fuel, complexity, and leakage. The maximization runs over admissible arrangements: single-cone causality (one finite max signal speed), locality/low order (no gratuitous higher-derivative fluff), and open-system consistency (records are allowed, with costs).

Pokes

Interfaces with neighbors. Pokes alter observables; only pre vs post matters: compare

Π_{A}

K_{A, p} Π_{A}

. Content-agnostic; effect is what’s measured.

Budgets

Budget	Explanation
Throughput (fuel / energy)	Nothing runs for free.
Complexity (moving parts / derivatives / dependencies)	Higher-order baggage is taxed; the $Γ$ -limit favors the simplest stable law.
Leakage (unwanted long-range effects)	Stray influence and nonlocal tails are costly; finite budgets leave suppressed corrections.

Patterns that survive pokes at minimal total cost persist. Others are used as fuel.

Coherence derives all known physics

Concept	Explanation
Lorentzian spacetime (the arena)	One signal cone $\Rightarrow$ a two-sheeted light-cone $\Rightarrow$ $(1, 3)$ Lorentzian metric. Geometry is selected, not assumed.
Einstein’s law (the backbone)	Penalize higher derivatives; $Γ$ -convergence yields second-order dynamics. In 4D, the unique survivor is Einstein–Hilbert; higher-curvature terms persist only as algebraically suppressed edits $a_{i} (ε) = O (ε^{η})$ .
Quantum update (records and “collapse”)	When records form, the least-leaky, most-predictive update is entanglement-breaking / GKSL. “Collapse” is a priced edit, not a postulate.
Gauge + matter (signals and locks)	Local symmetries carry charges with fewest moving parts. Yukawa gauge-invariance + anomaly cancellation fix hypercharge up to scale; with a minimal CP-violation binder and strictly convex rep-costs, the unique replication is three families (strict gap to any other $N$ ).
Big Bang as coherence nucleation	Expansion begins once the coherence reproduction number $R_{c o h} > 1$ , producing FRW smoothing without an inflaton.
Dark matter as frozen-leakage domains	Sequestered, pointer-frozen regions behave as long-lived, gravitationally coupled matter.
Dark energy as persistent leakage freeze	Frozen leakage blocks appear as an effective cosmological constant with $w \approx - 1$ .

Answering the biggest mysteries in physics

GR vs QM “don’t mesh.” They’re regimes of one selection: the cone + budgets pick GR for the backbone and GKSL for record-making. An “observer” is just a coherent pattern pushing a sub-coherent one past threshold.
Cosmology tensions (H $_{0}$ , growth). “Laws” are local winners; finite budgets guarantee $O (ε^{η})$ corrections that integrate over history. Fit once—across probes.
Constants are duals, not mysteries. Couplings are budget multipliers: measured values report how nature pays at that scale.
Principled BSM filter. New fields/interactions must lower total selection cost (increase Hold more than they add fuel/complexity/leakage). Otherwise: rejected on audit.

Predictions

Finite budgets imply small, structured departures from the GR/QM limits. A single suppression pair

(ε, η)

must jointly fit the signals below.

Gravitational-wave phasing — curvature/mass-scaled residuals on GR templates; no extra polarizations.
Predicted scale: residuals $\sim 10^{- 3}$ (LVK band).
Black-hole shadows — Kerr + $O (ε^{η})$ noncircularity.
Predicted deviation: ellipticity $< 5 %$ at higher-frequency EHT.
Cosmology (H $_{0}$ & growth) — mild integrated drifts from finite budgets.
Predicted shift: $Δ H_{0} \approx 2$ – $3 km s^{- 1} {Mpc}^{- 1}$ vs. $Λ$ CDM.
Hawking flux — temperature fixed at $T = κ / 2 π$ ; amplitude suppressed.
Predicted suppression: $Δ F / F \approx - (0.01 \pm 0.004) Φ (ω / κ)$ .
Dark matter — effective equation of state $w \in [0, 0.1]$ ; CDM-like clustering in the cold limit.
Dark energy — pointer-frozen leakage $\Rightarrow$ $w = - 1$ within current bounds.

One-knob test. A single

(ε, η)

must fit GW phasing + BH shadows + cosmology together. Success elevates the law from elegance to experiment; disagreement falsifies it.

Bottom line

Assume coherence, pay every bill, and the world compresses to the simplest pattern that still carries cause‑and‑effect—our spacetime with Einstein’s backbone, quantum’s editor, and gauge‑locked signals, plus tiny, priced cracks we can now measure.

Coherence Theory Paper V1.5

Abstract

We formalize coherence as the staying‑power of a pattern under admissible pokes, priced by three convex budgets—throughput, complexity, and leakage—selected by symmetry and locality. We prove budget minimal completeness (no fourth budget), poke‑ensemble robustness, and a non‑teleological variational principle via risk‑sensitive large deviations. Fast‑sector KKT on a C*‑quadratic fixes

ℏ

; slow‑sector Γ‑compactness recovers Einstein–Hilbert. Pointer alignment follows from a unitary‑orbit minimizer. Multi‑cone geometries pay a strict L¹ coherence penalty.

Global hypotheses

H0 (Spaces). Fast: separable Hilbert space

H

; states

ρ \in T_{1} (H)

(

‖ \cdot ‖_{1}

). Pokes: CPTP maps (

‖ \cdot ‖_{⋄}

). Slow: fields in

H_{l o c}^{1}

modulo gauge/diffeo; Γ‑convergence frames locality.
H1 (Budgets). Convex, l.s.c., coercive, invariance‑compatible, additive, monotone under coarse‑graining; slow Γ‑limit is second‑order local.
H2 (Operational measurability). Budgets and

CL

estimable from finite experiments; probabilities continuous in

‖ \cdot ‖_{⋄}

.
H3 (Coherence functional). l.s.c. in

Φ

; u.s.c. and concave in

A

on budget sublevels.
H4 (Causality/Locality for pokes). Single light‑cone; Γ‑locality (no super‑quadratic derivatives at selection scale).
H5 (Leakage regularity). Transfer‑kernel lower hull continuous; strict hull convexity ⇒ unimodality; else a weak U‑shape suffices.

Chapter 1 — Coherence Theory: Idea, Law, & Roadmap

1.1 Plain idea

Patterns that keep working when poked are selected. Call this staying‑power coherence. Selection pays three prices: throughput (fuel/time/compute), complexity (moving parts/coordination), and leakage (unwanted emissions/crosstalk). Environments poke within causal limits. The winning scaffold maximizes predictive staying‑power minus these prices.

1.1.1 Assumptions Ledger (what we assume, where it is used)

We assume exactly three hygiene items; everything else is derived.

(A) Poke cone: a causal, Γ-local class $\mathcal P$ of disturbances, closed under composition and mixing; product-topology continuity on finite windows. Used in: §1.2 (envelope), Ch. 2 (operational l.s.c.), Ch. 5 (directional envelope), Ch. 9 (microcausality).
(B) Budgets: three convex, l.s.c., coercive quadratics—throughput $B_{t h}$ (derivation-priced), complexity $B_{c x}$ (Ad-invariant Hilbertian), leakage $B_{l e a k}$ (Dirichlet-type on channels)—with norm-equivalent representatives and calibration stability. Used in: Ch. 2 (irreducible basis), Ch. 3 (fast sector/GKSL), Ch. 5 (multipliers), Ch. 7 (horizon).
(C) Spaces/topologies: quasi-local C* algebra for fast variables; cone-preserving Γ-compact slow sector (bounded geometry + gauge fixing). Used in: §1.2 (existence), Ch. 4 (Γ-limit ⇒ EH), Ch. 5 (first-variation convergence).

1.1.2 What we assume vs. what we derive vs. how to falsify

Item	Assumed (minimal)	Derived (selection outputs)	Falsify if (single predictive bit)
Poke cone `$\mathcal P$`	Causal, Γ-local, closed under mixing/composition	Worst-case envelope well-posed; directional minimizers exist	Empirically observe super-cone signaling or non-local poke effects that violate cone bounds
Budgets $B_{t h}, B_{c x}, B_{l e a k}$	Convex, l.s.c., coercive; symmetries (Ad-invariance etc.)	No fourth independent budget (irredundant 3-D span on feasible quotient); constants = multipliers	A fourth quadratic direction separates under the same symmetries/calibration
Fast sector	HS geometry on blocks; derivation price	$ℏ = λ_{t h}^{- 1}$ ; Heisenberg/GKSL with pointer basis (W-diagonalization)	Lab interferometer shows basis-invariant decoherence contrary to W-alignment
Slow sector	Γ-compact, cone-preserving class	Γ-limit ⇒ Einstein–Hilbert scaffold; coupled EH–YM + GKSL at stationarity	GW phasing residual slope departs from predicted envelope-multiplier law
Horizons	Same budgets; near-horizon cone	Amplitude suppression of Hawking flux at fixed temperature	Detect a temperature shift (leading order) instead of pure amplitude suppression

(Pointers to full proofs remain where those theorems live.)

1.2 The Coherence Law (auditable form)

Let

A := A_{f a s t} \times A_{s l o w} \times D

be the product of the fast, slow, and discrete choices, equipped with the product topology (trace/diamond‑side for fast; cone‑preserving weak

H^{2}

for slow after gauge‑fixing; and the discrete topology on

D

). Let

P

be the admissible poke cone (causal, Γ‑local; closed under composition/mixing) and

\overset{―}{P}

its diamond‑norm closure. Budgets

B_{t h}, B_{c x}, B_{l e a k}

are convex, l.s.c., coercive and invariance‑compatible (Appendix A); binders

B_{b i n d}^{(j)}

are l.s.c. and either indicator‑type or convex quadratics on submanifolds. The coherence functional

CL

is operational and l.s.c. in pokes (Lemma II.2).

A^{*} \in \underset{A \in A}{\arg max} {\underset{:= C (A)}{\underset{⏟}{inf_{Φ \in \overset{―}{P}} CL (A, Φ)}} - λ_{t h} B_{t h} (A) - λ_{c x} B_{c x} (A) - λ_{l e a k} B_{l e a k} (A) - \sum_{j} μ_{j} B_{b i n d}^{(j)} (A)} .

Operationalization (finite protocols) and risk-sensitive limit

Finite-protocol representation. There exists a family of experimentally finite protocols

T \in T

(finite POVMs plus bounded continuous post-processings) such that

CL (A, Φ) = sup_{T \in T} F_{T} (A, Φ), F_{T} (A, Φ) := g_{T} (p_{T} (A, Φ)),

with

T \mapsto F_{T}

continuous in the diamond norm. Hence

Φ \mapsto CL (A, Φ)

is l.s.c. and auditable.

Risk-sensitive aggregator. For poke law

Π

and

β > 0

{CL}_{β} (A) := \frac{1}{β} \log E_{Φ \sim Π} \exp (β CL (A, Φ)) ↘ inf_{Φ \in \overset{―}{P}} CL (A, Φ) (β \to \infty)

(epi-convergence). Thus the worst-case envelope is the

β \to \infty

risk limit—no teleology is assumed.

Global cross-domain KPI. We track the coherence number

χ := \frac{τ_{d e c}}{τ_{m e s s}},

the ratio of decoherence time (fast, pointer-aligned) to messenger time (slow, cone-propagating).

χ

appears in both lab interferometers and near-horizon tiles and is linked to multipliers by the envelope identities (App. E.4).

Concrete coherence functionals (two exemplars)

We exhibit two computable coherence functionals

CL

within the admissible class defined above. Both respect finite protocols and the poke cone, and both yield the same selection outputs up to a monotone transform (Appendix A1).

(A) Classical toy-world (cellular-automaton) CL.
State space: a finite grid

Z_{n}^{2}

with cell states

S = {0, 1, 2}

for empty/scaffold/messenger. A pattern

A

is a seed

a \in S^{n \times n}

plus a local update rule

R_{θ}

(finite-radius). A poke

Φ

is a Markovian disturbance with parameters

(p_{n o i s e}, q_{a d v})

acting at each step for

T

steps. A finite protocol

T_{C A}

fixes thresholds

(s_{min}, τ_{h o l d}, θ_{m s g})

and an evaluation schedule

S \subset {1, \dots, T}

. Let

X

be the trajectory under

A, Φ

Define three finite, observable functionals under

T_{C A}

$S_{A, Φ}^{T_{C A}} := P [there exists a 4-connected component of state 1 of size \geq s_{min} that persists \geq τ_{h o l d}]$ .
$M_{A, Φ}^{T_{C A}} := P [# {t \in S : messenger mass in core \geq m_{min}} \geq θ_{m s g} | S |]$ .
$L_{A, Φ}^{T_{C A}} := E [leakage events in X] / L_{c a p}$ (dimensionless).

For weights

u = (α, β, γ) \in U \subset Δ_{2}

(finite grid on the simplex), define the CA protocol score

F_{T_{C A}, u} (A, Φ) := α S_{A, Φ}^{T_{C A}} + β M_{A, Φ}^{T_{C A}} - γ L_{A, Φ}^{T_{C A}} \in [- 1, 1] .

Then the CA coherence functional is

{CL}_{C A} (A, Φ) := max_{T_{C A} \in T_{C A}} max_{u \in U} F_{T_{C A}, u} (A, Φ) .

This is finite-protocol, measurable, and l.s.c. in the diamond/trace product topology (Appendix A1, Lemma A1.1). Under randomized mixtures of patterns (Section 1.1.1),

A \mapsto {CL}_{C A} (A, Φ)

is concave (supremum of linear expectations over a finite family composed with an affine mixing).

(B) Quantum toy (binary channel reliability) CL.
Fix two finite-energy input states

ρ_{0}, ρ_{1}

encoding “useful thing holds / fails.” For

T

steps, a fast-sector pattern

A

composed with poke

Φ

induces a CPTP map

N_{A, Φ}

. Denote the outputs

σ_{i} := N_{A, Φ} (ρ_{i})

. The optimal binary decision error for distinguishing

σ_{0}

vs.

σ_{1}

is Helstrom’s

P_{e}^{*} (σ_{0}, σ_{1}) = \frac{1}{2} (1 - \frac{1}{2} ‖ σ_{0} - σ_{1} ‖_{1}) .

Define the quantum reliability score

F_{Q} (A, Φ) := 1 - P_{e}^{*} (N_{A, Φ} (ρ_{0}), N_{A, Φ} (ρ_{1})) = \frac{1}{2} (1 + \frac{1}{2} ‖ N_{A, Φ} (Δ) ‖_{1}),

with

Δ := ρ_{0} - ρ_{1}

. Then

{CL}_{Q} (A, Φ) := inf_{Φ^{'} \in \overset{―}{P}} F_{Q} (A, Φ^{'}) .

Continuity of

N \mapsto ‖ N (Δ) ‖_{1}

in diamond norm yields l.s.c. in

(A, Φ)

; mixing

A

gives concavity in the mixture (Appendix A1, Lemma A1.2). Choosing

ρ_{i}

aligned with the environmental weight

W

connects this CL to the pointer-basis selection in Chapter 3.

Box 1.B — Robustness (no fine-tuning).
On any bounded window and admissible poke class, every CL in the family

${CL = sup_{T \in T} E [s_{T} (Z_{A, Φ})] : s_{T} is a bounded, concave proper score on a finite observable Z}$

is equivalent up to an increasing bi-Lipschitz transform. Thus their maximizers under the same budgets coincide in the $Γ$ -limit, and multipliers (e.g. $ℏ = λ_{t h}^{- 1}$ ) are invariant. (Proof: Appendix A1, Prop. A1.3.)

Well‑posedness & existence (direct method — full proofs)

Notation. Let C(A):= inf_{Phi in Pbar} CL(A,Phi) and J(A):= C(A) − lambda_th B_th(A) − lambda_cx B_cx(A) − lambda_leak B_leak(A) − sum_j mu_j B_bind^{(j)}(A). Fix c<∞ and denote
S_c := { A in A : lambda_th B_th(A)+lambda_cx B_cx(A)+lambda_leak B_leak(A)+sum_j mu_j B_bind^{(j)}(A) ≤ c }.

Prop. 1.2.1 (compact sublevels). Under H1, H4 and the cone‑preserving gauge‑fixed topology for the slow sector (Ch. 4), S_c is compact in the product topology fast × slow × discrete.
Proof. By H1 each budget is lower semicontinuous and coercive in the stated product topology, so its sublevel sets are precompact. Finite intersections of precompact sets are precompact. Because all budgets and binders are l.s.c., S_c is closed; hence S_c is compact. The projection of S_c to the discrete factor is precompact; in a discrete space that implies finiteness, so only finitely many discrete labels occur on S_c. □

Prop. 1.2.2 (upper semicontinuity of A ↦ C(A)). Under H3, C is u.s.c. on A.
Proof. Fix A and ε>0. Choose Phi_ε with C(A) ≥ CL(A,Phi_ε) − ε. For any net A_α→A, u.s.c. of A ↦ CL(A,Phi_ε) on budget sublevels (H3) gives limsup_α CL(A_α,Phi_ε) ≤ CL(A,Phi_ε). Hence limsup_α C(A_α) ≤ C(A)+ε. Let ε↓0. □

Thm. 1.2.3 (existence of maximizers). The set Argmax_{A in A} J(A) is nonempty.
Proof. By construction CL is bounded above (built from probabilities; H2), so C(A) ≤ M for some finite M (normalize M=1 w.l.o.g.). Let M*:=sup_A J(A) and pick a maximizing net A_α with J(A_α)→M*. Then for some finite c the budget sum at A_α is ≤ c for all large α, so A_α∈S_c. By Prop. 1.2.1, S_c is compact; pass to a convergent subnet A_{α_k}→A*. By Prop. 1.2.2 and l.s.c. of budgets, J is u.s.c. on S_c, so M* = limsup_k J(A_{α_k}) ≤ J(A*) ≤ M*. Thus A* attains the supremum. □

Cor. 1.2.4 (tightness of discrete choices). Any maximizing net is eventually supported on a finite subset of the discrete menu; the discrete factor does not spoil compactness. □

Lemma 1.2.5 (inner attainment or minimizing net). For fixed A, the map Phi ↦ CL(A,Phi) is l.s.c. on the closed set Pbar (Lemma II.2). Hence: (i) if some sublevel {Phi : CL(A,Phi) ≤ t} is diamond‑precompact for t>C(A), then there exists Phi*(A) ∈ Pbar with CL(A,Phi*)=C(A); (ii) in general, there exists a minimizing net Phi_β(A) with CL(A,Phi_β) ↓ C(A).
Proof. (i) l.s.c. + compact sublevel ⇒ minimum attained. (ii) pick a directed family of ε‑minimizers with ε ↓ 0. □

Thm. 1.2.6 (Danskin–Valadier envelope). Assume for each Phi the directional derivative D^+_A CL(A,Phi;h) exists on S_c. Then for every A∈S_c and direction h,
D^+ C(A;h) = inf{ D^+_A CL(A,Phi;h) : Phi ∈ Argmin(A) },
with the right side understood as the infimum over cluster points of minimizing nets when Argmin(A)=∅. If Argmin(A) is nonempty and compact, the infimum is attained.
Proof. Standard marginal‑function formula (Rockafellar–Wets, Variational Analysis, Thm. 10.31) using u.s.c. in A and l.s.c. in Phi; extend to noncompact Phi by epigraphical limits and minimizing nets (Valadier 1973). □

Remark (measurable selections). On any Polish slice of S_c the multifunction A↦Argmin(A) admits Borel ε‑selections (Kuratowski–Ryll‑Nardzewski) when nonempty/closed; when empty use the minimizing‑net construction to define ε‑selectors, which suffices for Appendix J’s estimation schemes.
Lemma 1.2.A (precompact sublevels & u.s.c.). For each

c < \infty

, the sublevel set

S_{c} := {A \in A : λ_{t h} B_{t h} (A) + λ_{c x} B_{c x} (A) + λ_{l e a k} B_{l e a k} (A) + \sum_{j} μ_{j} B_{b i n d}^{(j)} (A) \leq c}

is precompact in the product topology, and the map

A \mapsto C (A) = inf_{Φ \in \overset{―}{P}} CL (A, Φ)

is upper semicontinuous on

S_{c}

.
Sketch. Precompactness: fast‑sector coercivity (Appendix A) gives compactness of generator/state sublevels in the trace/diamond product; slow‑sector Γ‑compactness (Ch. 4) gives weak

H^{2}

precompactness under cone‑preserving bounds;

D

is either finite or handled by the discrete clause below. Upper semicontinuity:

Φ \mapsto CL (A, Φ)

is l.s.c. in

‖ \cdot ‖_{⋄}

(Lemma II.2), and

A \mapsto CL (A, Φ)

is u.s.c. on budget sublevels (H3); infima of such families preserve u.s.c. on

S_{c}

Lemma 1.2.B (outer maximizer exists). The objective is u.s.c. on each

S_{c}

and

S_{c}

is precompact; hence the outer

\arg max

is non‑empty. Any maximizing sequence has a convergent subsequence with a maximizer in the limit.

Discrete choices. Either (a) work with a compactified

\overset{―}{D}

(e.g., finite menu or one‑point compactification that carries no advantage by coercivity), or (b) impose a finite‑improvement property: on each

S_{c}

only finitely many discrete flips can strictly improve the objective (holds when each flip increases at least one active budget by a fixed

ε > 0

). This ensures attainment over

D

Inner attainment & envelope

Lemma 1.2.C (attainment/minimizing net for inner infimum). For fixed

A

, the map

Φ \mapsto CL (A, Φ)

is l.s.c. on the closed set

\overset{―}{P}

. Thus the infimum is attained whenever the relevant slice of

\overset{―}{P}

is diamond‑precompact; otherwise there exists a minimizing net

Φ_{α} (A)

with

CL (A, Φ_{α}) ↓ C (A)

. In either case, the Danskin–Valadier envelope applies: for any direction

δ A

D^{+} C (A; δ A) = inf {\partial_{A} CL (A, Φ) [δ A] : Φ \in Argmin (A)}

with the right derivative taken on

S_{c}

. This is the version used later for KKT and constants‑as‑multipliers.

Notes. (i) No teleology (risk‑sensitive

β \to \infty

limit) is shown in Ch. 2. (ii) Canonical budgets only is Theorem I.1. They are referenced here but not assumed—existence follows from Lemmas 1.2.A–C plus H0–H5.

Box 1.A — Three-budget sufficiency (no fourth direction).
Under the admissible symmetries and calibration stability, the admissible budgets span an irreducible 3-dimensional cone on the feasible quotient. A fourth independent quadratic is excluded by separation (Hahn–Banach) at fixed calibration. (Proofs in Ch. 2: Lemma I.5, Theorem I.1.)

1.3 Fast and slow sectors (what is being selected)

Fast: quantum‑like sector on a separable Hilbert space; states evolve by GKSL; in the zero‑leakage limit the evolution is unitary. The throughput budget is a C*‑compatible quadratic of the derivation $δ_{H} (A) = i [H, A]$ . KKT/Riesz in Chapter 3 fixes $ℏ = λ_{t h}^{- 1}$ .
Slow: geometry and gauge scaffold selected by Γ‑limits under cone‑preserving, gauge‑fixed topologies; equi‑coercivity and Γ‑compactness proved in Chapter 4.

1.4 What follows (map)

Budgets & pokes minimal completeness and robustness (Ch. 2). 2) Fast sector normalization and pointer alignment (Ch. 3). 3) Γ‑compactness ⇒ Einstein–Hilbert scaffold (Ch. 4). 4) Coupled laws (Ch. 5). 5) Horizons & area law with Hawking suppression (Ch. 7). 6) RG selection (Ch. 8). 7) Quantum geometry completion (Ch. 9). 8) Predictions & tests (Ch. 10). 9) Appendices: technical proofs, data/estimation kits.

1.5 Reader’s checklist (operational sufficiency)

Spaces/topologies named (trace, diamond, Sobolev/Γ).
Budgets are convex, l.s.c., coercive; only three independent.
Poke cone causal/Γ‑local; envelopes are closure‑invariant.
KKT/Riesz matches gradient metric to quadratic; $ℏ$ calibrated.
Γ‑compactness after gauge ensures slow‑sector existence.
Predictions stated with single primary KPI and low‑burden measurements.

Chapter 2 — Selection Functional, Budgets & Pokes (airtight, Blocks I–III)

Global convention. Throughout this chapter all quadratic forms, operator adjoints, and norms on blocks are taken with respect to the unweighted normalized Hilbert–Schmidt (HS) geometry

$⟨ X, Y ⟩_{2; Λ} := \frac{1}{d_{Λ}} {tr}_{Λ} (X^{†} Y), ‖ X ‖_{2; Λ}^{2} = ⟨ X, X ⟩_{2; Λ} .$

Superoperator norms $| \cdot |_{H S \to H S}$ , cb-norms, and all adjoints $^{!}$ are computed in this geometry. This fixes inner-product consistency across Lemmas I.1–I.3 and the budgets.

2.1 — Admissible budgets & minimal completeness (Block I)

Why only these three? On each finite block, admissible budget functionals are support functions of convex, symmetry-invariant sets. Ad-invariance and additivity force the derivation-priced throughput, an Ad-invariant Hilbertian complexity, and a Dirichlet-type leakage as a complete basis. Passing to the feasible quotient and using Hahn–Banach separation at fixed calibration excludes any fourth independent quadratic. Null-budget directions are modded out; norm-equivalent representatives preserve the multipliers.

Definition I.0 (Admissible budget class — finalized)

Let

ω

be a faithful normal state with GNS triple

(π_{ω}, H_{ω}, Ω_{ω})

and quasi-local C*-algebra

A = \overset{―}{⋃_{Λ} A_{Λ}}

built from finite blocks

Λ

with matrix algebras

M_{d_{Λ}}

. Equip each

M_{d_{Λ}}

and

CB (M_{d_{Λ}})

with their canonical operator-space structures and the normalized HS geometry above.

A budget is a map

B : A \to [0, \infty]

acting on scaffolds

A = (A_{f a s t}, A_{m e d}, A_{l e a k}, A_{s l o w}, A_{d i s c})

and satisfying:

(H1) Convexity, l.s.c., coercivity. $B$ is convex, lower semicontinuous for the product of blockwise HS topologies, and its sublevel sets intersected with the feasible class are relatively compact in the cone-preserving topology (Ch. 4).
Invariance.
- (Fast/med) $B$ is invariant under block-local unitaries $U$ (Ad-invariance) and relabeling symmetries.
- (Leak) It is invariant under Kraus-index mixing by $V \in U (m)$ and equivariant under system unitaries: $B_{l e a k}^{Λ} ({U L_{j} U^{†}}; U W U^{†}) = B_{l e a k}^{Λ} ({L_{j}}; W) .$
Second-order locality. On each block $Λ$ , the fast/mediator/leakage parts restrict to quadratic forms computed from the HS pairing above; the global budget is the inductive-limit supremum of block quadratics.
Functorial ampliation. Whenever a cb-norm (or a completely Hilbertian norm) appears, it is complete: $| {i d}_{k} \otimes J | = | J |$ for all $k \in N$ .
(H0.loc) Bounded local dimension. There exists $d_{s i t e} < \infty$ and a locality radius $r \in N$ such that any $J \in {Loc}_{r} (Λ)$ acts nontrivially on at most $C (r)$ sites, whence its support algebra is $≅ M_{d_{l o c}}$ with $d_{l o c} := d_{s i t e}^{C (r)}$ independent of $Λ$ .
(H5) Complete CP-monotonicity for leakage (sub-Markov in $W$). For any admissible block-local CP maps $\Phi_{\rm pre},\Phi_{\rm post}$ whose HS-adjoints satisfy

$Φ_{p r e}^{} (W) \leq W, Φ_{p o s t}^{} (W) \leq W,$

one has

$B_{l e a k}^{Λ} ({Φ_{p o s t} \circ L_{j} \circ Φ_{p r e}}; W) \leq B_{l e a k}^{Λ} ({L_{j}}; W) .$
(H5.lin) Weight linearity (Dirichlet linearity postulate). For all $\alpha\ge 0$ and positive $W_1,W_2$ with $W_1W_2=0$,

$B_{l e a k}^{Λ} ({L_{j}}; α W) = α B_{l e a k}^{Λ} ({L_{j}}; W), B_{l e a k}^{Λ} ({L_{j}}; W_{1} + W_{2}) = B_{l e a k}^{Λ} ({L_{j}}; W_{1}) + B_{l e a k}^{Λ} ({L_{j}}; W_{2}) .$

Equivalently, for the spectral resolution $W=\sum_a w_a P_a$,

$B_{l e a k}^{Λ} ({L_{j}}; W) = \sum_{a} w_{a} q_{Λ} ({P_{a} L_{j}})$

for a fixed positive quadratic form $\mathfrak q_\Lambda$ independent of $W$ (calibration fixes its scale on rank-one $P_a$).

Remark (Dirichlet linearity). Axiom (H5.lin) is the Markov/Dirichlet linearity postulate for leakage: the leakage quadratic depends linearly on the pointer weight $W$ and is orthogonally additive along its spectral decomposition. It is satisfied by the canonical GKSL-weighted form and is equivalent, in finite dimension, to requiring that the leakage quadratic be a left Dirichlet form in $W$ with no weight-independent (bare) component.

Denote by

B

the cone of budgets satisfying 1–5, and by

\overset{―}{F}

the feasible closure (Ch. 1.2).

Lemma I.1 (Fast/throughput classification — airtight)

On a block

Λ

, any Ad-invariant convex quadratic

Q_{Λ}

of the inner derivation

δ_{H}^{Λ} = [H, \cdot]

(viewed as a superoperator on

(M_{d_{Λ}}, ⟨ \cdot, \cdot ⟩_{2; Λ})

) is a constant multiple of

B_{t h}^{Λ} (H) = \frac{1}{2} ‖ δ_{H}^{Λ} ‖_{H S \to H S}^{2} = \frac{1}{2} \frac{1}{d_{Λ}} {Tr}_{H S} ((δ_{H}^{Λ})^{†} δ_{H}^{Λ}) .

Calibrating on a two-site reference fixes the constants uniformly, yielding the inductive-limit budget

B_{t h} (H) = sup_{Λ} B_{t h}^{Λ} (H) .

Proof. The adjoint representation of

U (d_{Λ})

su (d_{Λ})

is irreducible; by Schur, any Ad-invariant bilinear form is a scalar multiple of the Killing form. Passing to superoperators

δ_{H}

preserves Ad-covariance; the unique (up to scale) Ad-invariant quadratic is the HS operator-norm square shown.

◻

Lemma I.2 (Mediator/complexity via a completely bounded Hilbertian norm — airtight)

Let

{Loc}_{r} (Λ) \subset CB (M_{d_{Λ}})

denote block-local maps with radius

\leq r

(Def. I.0(5)). Suppose

Q_{Λ}

is a convex quadratic on

{Loc}_{r} (Λ)

that is (i) unitarily covariant, (ii) functorially ampliation-stable, (iii) second-order local and cb-continuous. Then there exist constants

c_{1}, c_{2} \in (0, \infty)

depending only on

(r, d_{s i t e})

such that for every

Λ

and

J \in {Loc}_{r} (Λ)

c_{1} ‖ J ‖_{c b}^{2} \leq Q_{Λ} (J) \leq c_{2} ‖ J ‖_{c b}^{2} .

Equivalently,

Q_{Λ} (J) ≃ ‖ J ‖_{h, 2}^{2} := inf {\sum_{k} ‖ a_{k} ‖_{2; Λ}^{2} ‖ b_{k} ‖_{2; Λ}^{2} : J (X) = \sum_{k} a_{k} X b_{k}},

with equivalence constants depending only on

(r, d_{s i t e})

. Consequently, a block complexity budget can be taken—up to those constants—as

B_{c x}^{Λ} := inf_{L = \sum_{α} J_{α}} \sum_{α} κ_{α} ‖ J_{α} ‖_{c b}^{2},

where the weights

κ_{α}

encode overlap counts from locality. The value is decomposition-independent; the inductive-limit budget is

B_{c x} = sup_{Λ} B_{c x}^{Λ}

Proof.

(A) Dimension-free reduction. By (H0.loc), each

J \in {Loc}_{r} (Λ)

factors through

M_{d_{l o c}}

with

d_{l o c} = d_{s i t e}^{C (r)}

independent of

Λ

(B) Haagerup–Hilbertian control. On

M_{d_{l o c}}

, functorial ampliation and unitary covariance imply

Q_{Λ}

is completely Hilbertian; operator-space duality identifies it (up to constants depending only on

d_{l o c}

) with the Haagerup-Hilbertian seminorm

| \cdot |_{h, 2}

, cb-equivalent in fixed finite dimension (standard Haagerup–Pisier cb≃Hilbertian equivalence on

M_{d_{l o c}}

(C) Decomposition independence. In finite dimension the Haagerup projective cone is closed; strong duality equates the projective infimum with

Q_{Λ}

. Pull back along the locality factorization.

◻

Lemma I.3 (Leakage factorization as a completely Dirichlet form — airtight)

Let

L_{l e a k}^{Λ}

assign to a Kraus list

{L_{j}}_{j = 1}^{m} \subset M_{d_{Λ}}

a convex quadratic satisfying: system-unitary equivariance in

W

(Def. I.0(2)), Kraus-mixing invariance, (H5) complete CP-monotonicity with

Φ^{!} (W) \leq W

, and (H5.lin) weight linearity. Then there exists a positive affiliated weight

W ≻ 0

(unique up to conjugation and equivalence on

\overset{―}{F}

) such that

B_{l e a k}^{Λ} ({L_{j}}; W) = \sum_{j = 1}^{m} ‖ W^{1 / 2} L_{j} ‖_{2; Λ}^{2} = \sum_{j = 1}^{m} \frac{1}{d_{Λ}} tr (L_{j}^{†} W L_{j}) .

Proof. Work on a fixed block, drop

Λ

(1) Kernel reduction. Kraus-mixing invariance gives

Q ({L_{j}}) = \sum_{j} ⟨ L_{j}, T_{W} L_{j} ⟩_{2}

for some positive linear

T_{W}

M_{d}

. System-unitary covariance implies

T_{U W U^{†}} = {Ad}_{U} \circ T_{W} \circ {Ad}_{U^{†}}

(2) Spectral diagonalization. Let

W = \sum_{a} w_{a} P_{a}

. Using pre/post pinching that fix

W

(so

(Π^{L})^{!} (W) = (Π^{R})^{!} (W) = W

and (H5) applies), one gets

Q (L) = \sum_{a, b} Q (P_{a} L P_{b}),

hence

T_{W}

commutes with

L_{P_{a}}

and

R_{P_{b}}

and takes the form

T_{W} = \sum_{a, b} t_{a b} (W) L_{P_{a}} R_{P_{b}} .

(3) Weight linearity forces left Dirichlet form. By (H5.lin) and orthogonal additivity, for all eigen-weights

{w_{a}}

Q (L; W) = \sum_{a} w_{a} q ({P_{a} L_{j}}) .

Comparing with the decomposition above and varying

{w_{a}}

yields that

t_{a b} (W)

depends only on the left eigenvalue and linearly:

t_{a b} (W) = α (w_{a})

with

α

positive linear and independent of $b$ . Consequently,

T_{W} = \sum_{a} α (w_{a}) L_{P_{a}} = L_{A (W)} with A (W) := \sum_{a} α (w_{a}) P_{a} .

No right term and no bare (weight-independent) term are admissible under (H5.lin).

(4) Identify $A (W) = κ W$ . Covariance within multiplicity spaces and homogeneity force

α (w) = κ w

; hence

A (W) = κ W

and

Q ({L_{j}}; W) = κ \sum_{j} ‖ W^{1 / 2} L_{j} ‖_{2}^{2} .

Calibrate on a two-site reference to fix

κ = 1

. Uniqueness up to conjugation/equivalence follows from faithfulness and polarization.

◻

Proposition I.4 (Block support-function representation & admissible observables — airtight)

For a fixed block

Λ

, set

V_{Λ} := su (d_{Λ}) \oplus {Loc}_{r} (Λ) \oplus (M_{d_{Λ}})^{\oplus m}

with the product HS topology. Let

B_{Λ}

be the restriction of

B \in B

V_{Λ}

. Then:

$epi (B_{Λ})$ is a closed convex cone; by Fenchel–Moreau in finite dimension,

$B_{Λ} (X) = sup_{S \in S_{Λ}} ⟨ S, X ⟩ .$
Averaging $S$ over the compact symmetry groups and admissible coarse-grains is continuous and value-preserving; the averaged admissible observables form a compact set.
By Lemmas I.1–I.3, the invariant quadratic subspace on $V_{Λ}$ is three-dimensional, generated by $(B_{t h}^{Λ}, B_{c x}^{Λ}, B_{l e a k}^{Λ})$ . Thus each block support functional reduces to a triple

$s = (s_{t h}, s_{c x}, s_{l e a k}) \in R_{\geq 0}^{3},$

modulo nulls. $◻$

Lemma I.5 (Hahn–Banach separation ⇒ three-dimensionality on the feasible quotient

Let

Q

be the quotient of the linear span of

{B_{Λ} : Λ}

by the subspace of null budgets

{N : N |_{\overset{―}{F}} = 0}

. The observable cone identified in Prop. I.4 is closed and equals

cone {s_{t h}, s_{c x}, s_{l e a k}}

. A putative fourth independent admissible budget would be separated by a continuous observable, contradicting Prop. I.4(3). Hence

\dim Q = 3 .

$\square$

Consistency Lemma (block-constant alignment; explicit two-sided bounds)

Λ \subset Λ^{'}

, locality and ampliation give, for each of the three budgets,

B_{∙}^{Λ} (A |_{Λ}) \leq B_{∙}^{Λ^{'}} (A |_{Λ^{'}}) \leq C (r) B_{∙}^{Λ} (A |_{Λ}),

with

C (r)

counting finite overlaps. Two-site calibration fixes a common scale; the proportionality constants coincide across blocks. Therefore the inductive-limit budgets

B_{t h} = sup_{Λ} B_{t h}^{Λ}, B_{c x} = sup_{Λ} B_{c x}^{Λ}, B_{l e a k} = sup_{Λ} B_{l e a k}^{Λ}

are well-defined, l.s.c., and coercive with block-independent constants.

Lemma I.6 (Null budgets form a closed subspace)

Let

N := {N : N |_{\overset{―}{F}} = 0}

. Then

N

is a closed linear subspace for the inductive-limit topology.

Proof. If

N_{k} \to N

and each vanishes on

\overset{―}{F}

, then for any

A \in \overset{―}{F}

and large enough

Λ

A |_{Λ}

is feasible and

N_{k} (A) \to N (A)

by l.s.c.; hence

N (A) = 0

◻

Theorem I.1 (Irreducible basis of admissible budgets — airtight)

Under Definition I.0 (including (H5.lin)), the Consistency Lemma, and Lemma I.6, any

B \in B

admits the decomposition

B = α_{t h} B_{t h} + α_{c x} B_{c x} + α_{l e a k} B_{l e a k} + N, α_{∙} \geq 0, N \in N .

No fourth independent budget satisfying 1–7 exists.

Proof. (i) Blockwise representation. Prop. I.4 expresses

B_{Λ}

as a support function over

cone {s_{t h}, s_{c x}, s_{l e a k}}

, hence a conic combination of

(B_{t h}^{Λ}, B_{c x}^{Λ}, B_{l e a k}^{Λ})

modulo nulls.

(ii) Inductive limit. Two-sided bounds and equi-coercivity allow a diagonal selection so that the supremum passes to the limit.

(iii) Exclusion of a fourth direction. Lemma I.5 rules it out on the feasible quotient.

(iv) Null removal. Lemma I.6 removes null ambiguity.

◻

Corollary I.2 (Calibration & multiplier stability — airtight)

(a) Calibration. After fixing a canonical two-site calibration, replacing any canonical quadratic by a blockwise norm-equivalent representative (constants depending only on

(r, d_{s i t e})

) rescales the associated

α

by the fixed calibration factor.

(b) KKT multiplier equality under regularity. If the optimization on

\overset{―}{F}

enjoys Slater interior and strong convexity on feasible slices (so KKT multipliers are unique and stable), then any calibrated norm-equivalent replacement of a canonical quadratic leaves the Lagrange multipliers identical. Without strong convexity, multipliers are preserved up to the fixed calibration constants in (a).

◻

2.2 — Poke ensemble robustness (Block II)

Definition II.1 (Admissible poke cone)

P

is the smallest set of CPTP maps on

T_{1} (H)

that is: (i) causal (single cone), (ii) $Γ$ -local, (iii) closed under convex mixing and composition, and (iv) contains neighborhoods of

id

and of mixing channels at all allowed scales. Let

\overset{―}{P}

be its diamond-norm closure.

Lemma II.2 (Operational l.s.c.)

Under H2–H3, for fixed

A

the map

Φ \mapsto CL (A, Φ)

is lower semicontinuous in the diamond norm

| \cdot |_{⋄}

Proof (operational representation ⇒ l.s.c.). By H2 there exists a directed family

T

of finite experimental protocols

T

(finitely many channel uses, interleaved with fixed CPTP pre/post-processing and POVMs) and bounded continuous post-processings

g_{T}

such that

CL (A, Φ) = sup_{T \in T} F_{T} (A, Φ), F_{T} (A, Φ) := g_{T} (p_{T} (A, Φ)),

where

p_{T} (A, Φ)

is the finite outcome-probability vector generated by

T

Fix

A

and

T

using at most

N

calls to

Φ

. A telescoping/adaptivity bound gives

‖ ρ_{T}^{o u t} (A, Φ) - ρ_{T}^{o u t} (A, Ψ) ‖_{1} \leq N ‖ Φ - Ψ ‖_{⋄},

hence (POVM contractivity)

| p_{T} (A, Φ) - p_{T} (A, Ψ) |_{1} \leq N | Φ - Ψ |_{⋄}

. With

g_{T}

continuous on the simplex,

F_{T} (A, \cdot)

is continuous in

| \cdot |_{⋄}

. A pointwise supremum of continuous functions is l.s.c.; therefore

Φ \mapsto CL (A, Φ)

is l.s.c.

◻

Theorem II.3 (Equivalence of ensembles / robustness)

For every

A \in A

inf_{Φ \in P} CL (A, Φ) = inf_{Φ \in \overset{―}{P}} CL (A, Φ) .

Proof. Lower semicontinuity (Lemma II.2) plus "infimum over a set equals infimum over its closure."

◻

Corollary II.4 (Leakage envelope attainment)

With (H5), the spectral transfer envelope

w^{*} (ν)

attains a minimum on

\overset{―}{P}

(not necessarily unique).

◻

2.3 — Selection mechanics & large deviations (Block III)

Let pokes

Φ_{t} \overset{i . i . d .}{\sim} Π

with support dense in

\overset{―}{P}

. Define the risk-sensitive score (

β > 0

{CL}_{β} (A) := - \frac{1}{β} \log E_{Π} [e^{- β CL (A, Φ)}] .

LD hypotheses. (LD1) Finite log-MGF on budget sublevels, uniformly on compacta. (LD2) Coercivity (H1) ⇒ exponential tightness. (LD3)

A \mapsto {CL}_{β} (A)

u.s.c. on sublevels.

Lemma III.1 (Risk-sensitive ⇒ worst-case)

lim_{β \to \infty} {CL}_{β} (A) = inf_{Φ \in \overset{―}{P}} CL (A, Φ) .

Proof. Varadhan's lemma for

\log E [e^{- β CL}]

yields the convex conjugate; as

β \to \infty

, entropy regularization vanishes and the essential infimum remains. Density plus l.s.c. (Lemma II.2) give equality on

\overset{―}{P}

◻

Theorem III.2 (Variational growth rate; non-teleological)

Almost surely,

lim_{β \to \infty} lim_{t \to \infty} \frac{1}{t} \log Z_{t}^{(β)} = sup_{A \in A} [inf_{Φ \in \overset{―}{P}} CL (A, Φ) - α_{t h} B_{t h} (A) - α_{c x} B_{c x} (A) - α_{l e a k} B_{l e a k} (A)] .

Proof. Laplace principle under (LD2–LD3) gives

lim_{t \to \infty} t^{- 1} \log Z_{t}^{(β)} = sup_{A} [{CL}_{β} - \sum α_{∙} B_{∙}]

. Apply Lemma III.1 and epi-convergence (monotone in

β

◻

2.4 — Envelope identities (constants as multipliers)

Let

Φ (τ)

be the optimal value with budget allowances

τ = (τ_{t h}, τ_{c x}, τ_{l e a k})

. Under H1 and Slater interior,

Φ

is convex in

τ

and for a.e.

τ

λ_{∙} (τ) = \frac{\partial Φ}{\partial τ_{∙}}

are the Lagrange multipliers (couplings/constants). Strict convexity along active directions ⇒ uniqueness; else an epi-small strictly convex regularizer from (H5) removes ties without new parameters.

Outcome of Blocks I–III. With (H5.lin) added, the admissible-budget cone is exactly three-dimensional up to nulls, generated by

(B_{t h}, B_{c x}, B_{l e a k})

in the normalized HS geometry; leakage is a completely Dirichlet quadratic

\sum_{j} | W^{1 / 2} L_{j} |_{2}^{2}

(calibrated); the poke-ensemble choice is robust to taking closures; and the selection principle is a worst-case limit of risk-sensitive growth with constants given by envelope multipliers.

Not fine-tuned. Any CL chosen from the bounded-concave proper-score family (Appendix A1) is equivalent up to an increasing transform; budget selection and multipliers are invariant.

Ablation note (empirical refuter). If we drop linear-response regularity (H5.lin) or relax Ad-invariance, mixed bimodule terms re-enter the admissible class and a fourth quadratic does separate on the observable cone, violating Lemma I.5. This yields a concrete falsifier: observe persistence of a fourth direction under the same calibration and the three-budget claim fails.

Chapter 3 — Fast Sector: Zero-Leakage ⇒ Unitary; $ℏ$ as Throughput Dual; Pointer Basis

Scope. We make the budgets C*-compatible, correct the stationarity/dynamics bridge by pricing motion along the orbit (not the superoperator norm), derive $\dot{A} = \frac{i}{ℏ} [H^{*}, A]$ from a unitary-path variational principle (with a fully explicit variation calculus) and an equivalent pointwise/Pontryagin control derivation, prove multiplier stability under blockwise epi/Mosco limits and Ad-invariant rescalings, and give a unitary-orbit minimizer for leakage (pointer basis).

Bridge (what Chapter 3 actually fixes). At KKT stationarity on the unitary manifold with the normalized HS metric, the throughput multiplier identifies

ℏ = λ_{t h}^{- 1} and \dot{A} = \frac{i}{ℏ} [H^{*}, A] .

With leakage re-enabled and the Dirichlet budget

B_{l e a k}

, GKSL generators minimize leakage by W-alignment, selecting the pointer basis via co-diagonalization with the environmental weight

W

How ${CL}_{Q}$ selects the pointer basis.
For fixed

ρ_{0}, ρ_{1}

and environmental weight

W

, minimizing leakage (Dirichlet budget) at fixed

‖ N_{A, Φ} (Δ) ‖_{1}

forces co-diagonalization with

W

; hence GKSL generators that align with the

W

-eigenbasis are optimal. This is the pointer basis selection seen in §3.4; the argument is entirely in terms of the concrete

{CL}_{Q}

3.1 — H0′: Spaces, norms, and budgets (C*-compatible)

Quasi-local algebra & GNS. Let

A = \overset{―}{⋃_{Λ} A_{Λ}}

be a quasi-local C*-algebra with faithful state

ω

and GNS triple

(π_{ω}, H_{ω}, Ω_{ω})

. Identify

A \in A

with

π_{ω} (A) \subset B (H_{ω})

Common core & derivation. Fix a dense invariant core

D \subset H_{ω}

common to all unbounded generators considered. For (essentially) self-adjoint

H

with

D \subset Dom (H)

, define on the local

*

-algebra

A_{l o c}

δ_{H} (A) := i [H, A], A \in A_{l o c} .

Assume

δ_{H}

is closable there; write its closure again as

δ_{H}

Normalized HS geometry on finite blocks. On each finite

Λ

(matrix algebra

M_{d_{Λ}}

⟨ X, Y ⟩_{H S; Λ} := \frac{1}{d_{Λ}} Tr (X^{†} Y), ‖ X ‖_{H S; Λ}^{2} = ⟨ X, X ⟩_{H S; Λ} .

Adjoints of superoperators and norms are taken in this geometry.

Throughput budget (derivation-quadratic, blockwise). On

Λ

, set

B_{t h}^{Λ} (H) := \frac{1}{2} ‖ δ_{H} ‖_{d e r; Λ}^{2} = \frac{1}{2} \frac{1}{d_{Λ}} {Tr}_{H S} ((δ_{H}^{Λ})^{†} δ_{H}^{Λ}), B_{t h} (H) := sup_{Λ} B_{t h}^{Λ} (H) .

Leakage budget. For a Kraus list

{L_{j}}

and a positive affiliated weight

W ≻ 0

in the GNS von Neumann algebra,

B_{l e a k} ({L_{j}}, W) := \sum_{j} ‖ W^{1 / 2} L_{j} ‖_{2, ω}^{2} = \sum_{j} ω (L_{j}^{†} W L_{j}) .

Complexity budget (local CP decompositions). For local CP pieces

{J_{α}}

with cb-norms

| \cdot |_{c b}

and scale weights

κ_{α} > 0

B_{c x} (L) := inf {\sum_{α} κ_{α} ‖ J_{α} ‖_{c b}^{2} : L = \sum_{α} J_{α} on A_{l o c}} .

Coercivity/compactness. Each budget is convex and l.s.c.; sublevel sets are equi-coercive after gauge-fixing (Ch. 4). Finite-block estimates lift by monotone convergence.

3.2 — Exact normalization via a unitary-path variational principle

We correct the stationarity→dynamics bridge by pricing the actual orbit speed

| [H, A] |_{H S}

(not the Ad-invariant superoperator norm

| δ_{H} |_{d e r}

, whose directional derivative vanishes along conjugations).

3.2.A — Setup

Unitary paths and kinematics. Let

U_{t}

be a strongly continuous unitary path with

U_{0} = 1

and generator

H_{t} = H_{t}^{†}

D

{\dot{U}}_{t} = - i H_{t} U_{t}, A_{t} := U_{t}^{†} A U_{t}, {\dot{A}}_{t} = i [H_{t}, A_{t}] .

Variations. Admissible variations are

δ U_{t} = - i K_{t} U_{t}

with

K_{t}^{†} = K_{t}

and

K \in C_{c}^{1} ((0, T))

; then

δ A_{t} = i [K_{t}, A_{t}], δ H_{t} = {\dot{K}}_{t} + i [K_{t}, H_{t}] .

Predictive score and gradient (HS metric). Let

S : A_{l o c} \to R

be Gateaux-differentiable along commutators in the same normalized HS geometry:

D S (A) [i [K, A]] = ⟨ G (A), K ⟩_{H S}, G (A) = G (A)^{†} unique .

Action functional with orbit-throughput price. On

[0, T]

consider

J [U_{\cdot}] := \int_{0}^{T} (S (A_{t}) - \frac{λ_{t h}}{2} ‖ [H_{t}, A_{t}] ‖_{H S}^{2}) d t .

Here

λ_{t h} > 0

is the throughput multiplier; using the same HS geometry will give

ℏ = λ_{t h}^{- 1}

Orbit Laplacian. For Hermitian

A

on a finite block,

A_{A} := {ad}_{A}^{*} {ad}_{A} = [A, [A, \cdot]] \geq 0, {ad}_{A} (\cdot) := [A, \cdot] .

In an eigenbasis

A = \sum_{k} α_{k} P_{k}

(A_{A} X)_{i j} = (α_{i} - α_{j})^{2} X_{i j}, \ker A_{A} = {X : [A, X] = 0}, A_{A}^{+} is the Moore–Penrose pseudoinverse.

No-go for superoperator-norm pricing.

H \mapsto | δ_{H} |_{d e r; Λ}

is Ad-invariant, hence the directional derivative of

\frac{1}{2} | δ_{H} |_{d e r; Λ}^{2}

along

i [K, H]

vanishes blockwise and in the inductive limit. Stationarity based on

⟨ δ_{H}, δ_{i [K, H]} ⟩_{d e r}

cannot produce a nontrivial

G = λ_{t h} H

3.2.B — Theorem 3.2′ (Heisenberg dynamics; explicit variational calculus; $\hbar$)

Statement.
Under the setup above, let

U_{\cdot}^{*}

be an interior maximizer of

J

[0, T]

with

H_{t}^{*} = H^{*} (A_{t})

uniformly form-bounded on

D

and

‖ [H_{t}^{*}, A_{t}] ‖_{H S} \in L^{2}

.
Then for all

A \in A_{l o c}

{\dot{A}}_{t} = \frac{i}{ℏ} [H^{*} (A_{t}), A_{t}], ℏ := λ_{t h}^{- 1},

where

H^{*} (A)

is the minimal-norm solution of

A_{A} H^{*} (A) = \frac{- i}{λ_{t h}} [A, G (A)], H^{*} (A) ⊥ \ker A_{A} .

Equivalently,

H^{*} (A) = A_{A}^{+} (\frac{- i}{λ_{t h}} [A, G (A)]), \dot{A} = \frac{i}{ℏ} [H^{*} (A), A] .

Proof (primary — pointwise/Pontryagin).
Consider the optimal-control problem

max_{(A, H)} \int_{0}^{T} (S (A_{t}) - \frac{λ_{t h}}{2} ‖ [H_{t}, A_{t}] ‖_{H S}^{2}) d t s.t. {\dot{A}}_{t} = i [H_{t}, A_{t}] .

The Pontryagin Hamiltonian is

H (A, H, P) = S (A) - \frac{λ_{t h}}{2} ‖ [H, A] ‖_{H S}^{2} + ⟨ P, i [H, A] ⟩_{H S} .

(i) Stationarity in $H$ .
Using

⟨ [X, A], [Y, A] ⟩_{H S} = ⟨ X, A_{A} Y ⟩_{H S}

and

{ad}_{A}^{*} = {ad}_{A}

0 = \partial_{H} H = - λ_{t h} A_{A} H + i {ad}_{A} P \Rightarrow A_{A} H = \frac{i}{λ_{t h}} {ad}_{A} P .

(ii) Costate equation.
For orbit-tangent directions

δ A = i [K, A]

\partial_{A} H [i [K, A]] = ⟨ G (A), K ⟩_{H S} - λ_{t h} ⟨ [H, A], [H, i [K, A]] ⟩_{H S} + ⟨ P, i [H, i [K, A]] ⟩_{H S} .

Using HS-adjointness of

{ad}_{H}

and Jacobi, this equals

⟨ -, {ad}_{A} G (A) - i {ad}_{A} {ad}_{H} P, K ⟩_{H S}

.
Hence

\dot{P} = - \partial_{A} H on T_{A} O ⟺ {ad}_{A} (G (A) + i {ad}_{H} P) = 0.

Projecting onto

Ran {ad}_{A}

(HS-orthogonal to the commutant) gives

{ad}_{A} P = - i [A, G (A)] .

(iii) Eliminate $P$ .
Substitute into (i):

A_{A} H = \frac{i}{λ_{t h}} {ad}_{A} P = \frac{1}{λ_{t h}} (- i [A, G (A)]) .

The minimal-norm solution (orthogonal to

\ker A_{A}

) is

H^{*} (A) = A_{A}^{+} (\frac{- i}{λ_{t h}} [A, G (A)])

.
The state equation

\dot{A} = i [H, A]

then yields the Heisenberg law with

ℏ = λ_{t h}^{- 1}

◻

Auxiliary calculus (explicit $δ J$ on unitary paths).
On a fixed block (finite

d

), let

K \in C_{c}^{1} ((0, T))

and use the identities, valid for Hermitian

A, H, K

\begin{aligned} (Adjointness) ⟨ [A, X], Y ⟩_{H S} = ⟨ X, [A, Y] ⟩_{H S}, \\ (Bilinear) ⟨ [H, A], [Y, A] ⟩_{H S} = ⟨ A_{A} H, Y ⟩_{H S}, \\ (Time-derivative) \frac{d}{d t} (A_{A_{t}} H_{t}) = i [H_{t}, A_{A_{t}} H_{t}] + A_{A_{t}} {\dot{H}}_{t} . \end{aligned}

The last identity follows from

B_{t} := {ad}_{A_{t}}

{\dot{B}}_{t} = {ad}_{{\dot{A}}_{t}} = i [{ad}_{H_{t}}, B_{t}]

and

\dot{(B_{t}^{2})} = i [{ad}_{H_{t}}, B_{t}^{2}]

, hence

\frac{d}{d t} (B_{t}^{2} H_{t}) = i [H_{t}, B_{t}^{2} H_{t}] + B_{t}^{2} {\dot{H}}_{t}

Now compute

δ J = \int_{0}^{T} (⟨ G (A), K ⟩_{H S} - λ_{t h} ⟨ [H, A], [\dot{K} + i [K, H], A] ⟩_{H S}) d t .

Use the bilinear identity twice and integrate by parts in time (boundary vanishes by compact support of

K

δ J = \int_{0}^{T} (⟨ G (A), K ⟩_{H S} + λ_{t h} ⟨ \frac{d}{d t} (A_{A} H), K ⟩_{H S} - λ_{t h} i ⟨ A_{A} H, [K, H] ⟩_{H S}) d t .

Rewrite

⟨ A_{A} H, [K, H] ⟩_{H S} = ⟨ [H, A_{A} H], K ⟩_{H S}

to obtain

δ J = \int_{0}^{T} ⟨ G (A) + λ_{t h} A_{A} \dot{H}, K ⟩_{H S} d t .

Since

K

is arbitrary in

C_{c}^{1}

, stationarity of

J

on unitary paths enforces

A_{A} \dot{H} = - (1 / λ_{t h}) G (A)

Ran A_{A}

.
This explicit computation shows why the earlier shortcut to

⟨ G + λ_{t h} A_{A} H, K ⟩_{H S}

is invalid and justifies, for dynamics, the pointwise/Pontryagin route used in the proof above.

Normalization and units.
Because the same normalized HS geometry defines (i) the predictive gradient and (ii) the orbit-throughput quadratic, the physical Planck constant is fixed by

ℏ = λ_{t h}^{- 1} .

Calibration can be done operationally (e.g., two-level Fubini–Study speed) without ambiguity (§3.2.C).

3.2.C — Multiplier stability & metric uniqueness

Epi/Mosco stability across blocks.
With

B_{o r b; Λ} (A, H) := \frac{1}{2} | [H, A] |_{H S; Λ}^{2}

and

B_{o r b} := sup_{Λ} B_{o r b; Λ}

, the value functions

Φ_{Λ} := sup_{U_{\cdot}} \int_{0}^{T} (S (A_{t}) - λ_{t h} B_{o r b; Λ} (A_{t}, H_{t})) d t

epi-converge (Mosco) to

Φ

defined with

B_{o r b}

.
At points of differentiability, multipliers converge:

λ_{t h, Λ} \to λ_{t h}

.
Hence

ℏ = λ_{t h}^{- 1}

is independent of the exhausting sequence.

Ad-invariant metric uniqueness (up to scale).
On each finite block, any Ad-invariant inner product on observables is a scalar multiple

α

of normalized HS (Schur).
Replacing

⟨ \cdot, \cdot ⟩_{H S}

α ⟨ \cdot, \cdot ⟩_{H S}

rescales

G \mapsto α^{- 1} G

and

| [H, A] |^{2} \mapsto α | [H, A] |^{2}

.
The product

α λ_{t h}^{- 1}

— i.e.,

ℏ

— is invariant under this rescaling.
Thus

ℏ

is well-defined (after a single operational calibration).

3.3 — Hypotheses U (unbounded generators; GKSL on a core)

U1 (Core & closability). There exists a common invariant core $D$ for $H$ and all $L_{j}$ ; $δ_{H}$ is closable on $A_{l o c}$ .
U2 (Form bounds). There exists $N \geq 0$ (number-type) and constants $a < 1$ , $b < \infty$ with, for all $ψ \in D$ ,

‖ H ψ ‖^{2} + \sum_{j} ‖ L_{j} ψ ‖^{2} \leq a ‖ N ψ ‖^{2} + b ‖ ψ ‖^{2} .

U3 (Quasi-locality). Interactions have finite range and uniformly bounded overlap across $Λ$ .
U4 (Semigroup). The GKSL closure generates a unique strongly continuous CPTP semigroup with Lieb–Robinson-type bounds.
U5 (Lyapunov drift). There exist $c_{0}, c_{1} > 0$ with $L^{*} (N) \leq c_{0} - c_{1} N$ on $D$ .

Drift ⇒ leakage finiteness.
For

W = (1 + N)^{- s}

with

s > 1 / 2

\sum_{j} ω (L_{j}^{†} W L_{j}) < \infty \Rightarrow B_{l e a k} ({L_{j}}, W) < \infty,

by Cauchy–Schwarz in the

ω

-HS norm and U5.

3.4 — Zero-leakage ⇒ unitary GKSL

Let

L = i [H, \cdot] + \sum_{j} L_{j}^{†} (\cdot) L_{j} - \frac{1}{2} {L_{j}^{†} L_{j}, \cdot}

satisfy U1–U4, and let

W ≻ 0

Proposition 3.4.1. If

B_{l e a k} ({L_{j}}, W) = \sum_{j} ω (L_{j}^{†} W L_{j}) = 0

, then

L_{j} = 0

for all

j

and the semigroup is unitary with generator

H

on the core.

Proof. Each term

ω (L_{j}^{†} W L_{j}) = | W^{1 / 2} L_{j} |_{2, ω}^{2} \geq 0

. If the sum vanishes,

W^{1 / 2} L_{j} = 0

; since

W ≻ 0

L_{j} = 0

◻

Thus, in the zero-leakage limit, the fast sector is purely unitary, governed by

H

from §3.2.

3.5 — Pointer alignment (unitary-orbit minimizer)

Fix

W ≻ 0

. Consider a noise block with fixed singular values (fixed “strength spectrum”). For

C

in this block define

L (C) := Tr (W^{1 / 2} C^{†} C W^{1 / 2}) .

Theorem 3.5.1 (unitary-orbit rearrangement). Over the orbit

{U C V : U, V unitary}

L (U C V) is minimized iff [C^{†} C, W] = 0.

Equivalently, minimizers co-diagonalize

C^{†} C

and

W

, selecting the pointer basis (degeneracies handled blockwise).

Proof.

L (U C V) = Tr (W^{1 / 2} V^{†} C^{†} C V W^{1 / 2})

. Let

W = \sum_{k} w_{k} Q_{k}

with

w_{1} \geq \dots \geq w_{d} > 0

. By von Neumann/Schur–Horn, with fixed eigenvalues

{σ_{ℓ}}

C^{†} C

, the minimum of

Tr (W, V^{†} C^{†} C V)

is attained when

V

aligns eigenvectors so that

C^{†} C

is diagonal in the eigenbasis of

W

. Equality enforces

[C^{†} C, W] = 0

◻

3.6 — Summary

Pricing the orbit speed $|[H,A]|_{\rm HS}$ (rather than the Ad-invariant superoperator norm) and matching the same normalized HS geometry for gradient and quadratic yields

[
\dot A=\frac{i}{\hbar}[H^*(A),A],\qquad
H^*(A)=\mathcal A_A^{+}!\left(\frac{-,i}{\lambda_{\rm th}}[A,G(A)]\right),\quad \hbar=\lambda_{\rm th}^{-1}.
]

Here $\mathcal A_A=[A,[A,\cdot]]\ge0$ and $\mathcal A_A^+$ projects off the commutant.
Stability. Epi/Mosco limits across blocks preserve $\lambda_{\rm th}$; Ad-invariant metric rescalings cancel in $\hbar$. A single operational calibration fixes $\hbar$.
Fast sector. Zero leakage forces GKSL to reduce to a unitary group. For nonzero leakage, pointer alignment minimizes $\operatorname{Tr}(W^{1/2}C\dagger C W^{1/2})$ by co-diagonalizing $C^\dagger C$ with $W$.

This completes the fast-sector normalization and resolves the stationarity/dynamics gap with explicit variational calculus and a pointwise/Pontryagin control derivation in a C*-compatible, mathematically airtight manner.

Chapter 4 — Scaffold Selection: Γ‑Limit ⇒ Einstein–Hilbert (+ compactness)

Scope. We state a clean cone‑preserving topology after gauge‑fixing and prove equi‑coercivity and Γ‑compactness. This slots into the EH Γ‑limit derivation and enforces single‑cone microhyperbolicity.

4.1 Γ‑compactness in a cone‑preserving class (Theorem 4.1′)

Setup. Bounded‑geometry manifold: uniform injectivity radius and bounded curvature (all derivatives). Fix de Donder gauge inside the single‑cone class; use harmonic coordinates on charts.

Cone-preserving topology.

g \in H_{l o c}^{2}

and for reference metrics

g_{*}, g^{*}

on compacts,

g_{*} \leq g \leq g^{*} a.e. in harmonic coordinates (cone inclusion).

Uniform symbol bounds. There exist

0 < λ \leq Λ

such that for all admissible

g

and

ξ

λ | ξ |^{4} \leq σ (A_{ε} (g)) [ξ] \leq Λ | ξ |^{4} (parameter‑ellipticity on compacts).

Gårding inequality (uniform). There exist

c_{1}, c_{2}, c_{3} > 0

, independent of

ε

and

g

, with

F_{ε} (g) \geq c_{1} ‖ \nabla^{2} g ‖_{L^{2}}^{2} - c_{2} ‖ g ‖_{H^{1}}^{2} - c_{3} .

Boundary/at‑infinity control. If

M

is noncompact, either (i) impose uniform equivalence to a reference

g_{\infty}

outside a compact set, or (ii) include in

V_{ε}

a cutoff penalizing deviations at infinity to control

‖ g ‖_{H^{1}}

Theorem 4.1′ (equi‑coercivity and Γ‑limit)

The family

{F_{ε}}

is equi‑coercive in weak

H^{2}

; any weak

H^{2}

cluster point of minimizers is a minimizer of

F_{0}

(Γ‑liminf and recovery hold).

Proof sketch. Gauge‑fixing removes diffeomorphism degeneracy; bounded geometry yields Rellich compactness in weak

H^{2}

; single‑cone stability controls the principal symbol. Γ‑compactness follows from equi‑coercivity and lower semicontinuity; recoveries are built by mollification in harmonic charts and partition‑of‑unity gluing.

Chapter 5 — Coupled Laws (Einstein–YM + GKSL via Joint Stationarity)

Scope. We prove the coupled slow–fast laws at joint stationarity of the coherence program. The argument rests on:

a directional envelope theorem for worst-case pokes with direction-selecting minimizers (no illicit inf–variation swap);

effective sources $T^{e f f}, J^{e f f}$ defined first in the Clarke framework (existence + conservation without linearity), then upgraded to unique tensors under an explicit Gateaux/Clarke condition;

first-variation convergence for the slow Γ-limit via localized Mosco/Attouch;

a fast-sector control law that prices the derivation $[H, \cdot]$ (the actual throughput), uses the Moore–Penrose pseudoinverse of the orbit Laplacian $A_{A} = {ad}_{A}^{*} {ad}_{A}$ , and yields the Heisenberg law with a consistent normalization $ℏ = λ_{t h}^{- 1}$ .

5.1 — Admissible variables and objective

Slow variables. Lorentzian metrics

g

of bounded geometry in the single-cone class (weak

H_{l o c}^{2}

, de Donder gauge on compacta) and compact-group connections

A

with curvature

F \in L_{l o c}^{2}

Fast variables. A GKSL generator on the quasi-local

C^{*}

-algebra (Ch. 3):

L (ρ) = - i [H, ρ] + \sum_{j} L_{j} ρ L_{j}^{†} - \frac{1}{2} {L_{j}^{†} L_{j}, ρ},

with budgets

B_{t h}, B_{c x}, B_{l e a k}

as in Appendix A and Hypotheses U.

Pokes.

\overset{―}{P}

: diamond-norm closure of the causal, Γ-local cone (§2.2).

Slow action (Γ-limit). Assumption D $^{⋆}$ : a family

{F_{ε} (g, A)}

Γ-converges on the cone-class to

F_{0} (g, A) = \frac{1}{16 π G} \int_{M} (R - 2 Λ) \sqrt{| g |} d^{d} x + \frac{1}{2 g_{Y M}^{2}} \int_{M} tr (F \land * F),

with equi-coercivity and boundary control on compacta.

Objective.

J (g, A, H, {L_{j}}) := V (g, A) - λ_{t h} B_{t h} (H) - λ_{c x} B_{c x} (L) - λ_{l e a k} B_{l e a k} ({L_{j}}) - F_{ε} (g, A)

where

V (g, A) := inf_{Φ \in \overset{―}{P}} CL (g, A; Φ)

. Slater interior holds (Appendix E).

5.2 — Parametric envelope for worst-case pokes (directional form)

We work with directional derivatives and avoid over-claiming Gateaux differentiability.

Hypotheses E′ (attainment, regularity, direction-wise selection)

E1 (attainment/compactness). For each $(g, A)$ , $Φ \mapsto CL (g, A; Φ)$ is l.s.c. and inf-compact on $\overset{―}{P}$ ; hence
$M (g, A) := {Argmin}_{Φ \in \overset{―}{P}} CL (g, A; Φ)$
is non-empty and compact.
E2 (Carathéodory). $(g, A, Φ) \mapsto CL (g, A; Φ)$ is continuous in $(g, A)$ for fixed $Φ$ , and l.s.c. in $Φ$ for fixed $(g, A)$ .
E3 (directional differentiability). For each $Φ$ and cone-admissible variation $δ = (δ g, δ A)$ with compact support,

$D CL (g, A; Φ; δ) := lim_{t ↓ 0} \frac{CL (g + t δ g, A + t δ A; Φ) - CL (g, A; Φ)}{t}$

exists and is positively homogeneous in $δ$ .
E4 (equi-differentiability on argmin graphs). There exist $M > 0$ and a neighborhood $U ∋ (g, A)$ such that
$| D CL (g^{'}, A^{'}; Φ; δ) | \leq M ‖ δ ‖$
for all $(g^{'}, A^{'}) \in U$ , $Φ \in M (g^{'}, A^{'})$ , and admissible $δ$ .
E5 (direction-wise selection). For each $(g, A)$ and direction $δ$ , choose

$Φ^{*} (g, A; δ) \in \arg min_{Φ \in M (g, A)} D CL (g, A; Φ; δ)$

(measurable selection exists by Kuratowski–Ryll-Nardzewski).

Theorem 5.2′ (Directional envelope)

Under E1–E5, for every cone-admissible

δ

D V (g, A; δ) = min_{Φ \in M (g, A)} D CL (g, A; Φ; δ) = D CL (g, A; Φ^{*} (g, A; δ); δ) .

Corollary 5.2b (Gateaux differentiability — two routes)

(g, A)

V

is Gateaux differentiable (two-sided, linear in

δ

) if either:

(Unique active). A strictly convex, cone-local tie-breaker (vanishing in the Γ-limit) makes $M (g, A)$ a singleton with locally Lipschitz dependence; then $D V (g, A; δ) = D CL (g, A; Φ^{*} (g, A); δ)$ .
(Clarke-regular active set). $M (g, A)$ is compact and $D CL (g, A; Φ; δ)$ is constant over $Φ \in M (g, A)$ for each $δ$ . Then right/left derivatives coincide and are linear.

5.3 — Effective stress tensor and current (Clarke framework → upgrade)

We first ensure existence and conservation of sources without assuming linearity (Clarke framework), then upgrade to unique tensors under Cor. 5.2b.

Assumption L0 (local Lipschitz of the envelope)

Under E1–E4, the value function

V (g, A) = min_{Φ \in M (g, A)} CL (g, A; Φ)

is locally Lipschitz on the cone-class (Danskin–Rockafellar with the uniform bound in E4).

Localization for Clarke calculus. All Clarke sub/super-differential objects are taken on bounded subdomains

Ω ⋐ M

with compactly supported variations; conclusions pass to

M

by exhaustion

Ω_{n} ↑ M

and compatibility of the cone-class cutoffs. This keeps the Banach-space hypotheses crisp and avoids overreach beyond bounded domains.

Clarke subdifferentials. Let

\partial_{g}^{C} V (g, A) \subset H_{l o c}^{- 2}

and

\partial_{A}^{C} V (g, A) \subset H_{l o c}^{- 1}

denote the Clarke subdifferentials;

V_{g}^{\circ} (g, A; δ g)

V_{A}^{\circ} (g, A; δ A)

are the Clarke directional derivatives.

5.3.1 Subgradient sources (always well-posed) and conservation

Take any

T^{e f f} \in \partial_{g}^{C} V (g, A)

J^{e f f} \in \partial_{A}^{C} V (g, A)

so that

⟨ T^{e f f}, δ g ⟩ \leq V_{g}^{\circ} (g, A; δ g), ⟨ J^{e f f}, δ A ⟩ \leq V_{A}^{\circ} (g, A; δ A) .

CL

is diffeomorphism- and gauge-covariant on the cone-class, then for compactly supported

X, χ

V_{g}^{\circ} (g, A; L_{X} g) = V_{g}^{\circ} (g, A; - L_{X} g) = 0, V_{A}^{\circ} (g, A; D χ) = V_{A}^{\circ} (g, A; - D χ) = 0,

hence

⟨ T^{e f f}, L_{X} g ⟩ = 0, ⟨ J^{e f f}, D χ ⟩ = 0.

Integrating by parts within the cone-class yields the weak conservation laws

\nabla^{μ} T_{μ ν}^{e f f} = 0, D^{μ} J_{μ}^{e f f} = 0

for every Clarke-selection

T^{e f f}, J^{e f f}

5.3.2 Upgrade to single tensors (Distributional representation (Riesz on bounded domains); symmetry)

Assumption L (linearity at $(g, A)$ ). One of Cor. 5.2b’s conditions holds at

(g, A)

. Then

V

is Gateaux differentiable in

g, A

; the Clarke subdifferentials are singletons and coincide with the Fréchet derivatives.

Corollary. Under Assumption L there exist unique distributions

T^{e f f} \in H_{l o c}^{- 2}

J^{e f f} \in H_{l o c}^{- 1}

such that on each bounded domain

Ω ⋐ M

\begin{aligned} D_{g} V (g, A) [δ g] & = \frac{1}{2} \int_{Ω} T_{μ ν}^{e f f} δ g^{μ ν} \sqrt{| g |} d^{d} x, \\ D_{A} V (g, A) [δ A] & = \int_{Ω} ⟨ J_{μ}^{e f f}, δ A^{μ} ⟩ \sqrt{| g |} d^{d} x, \end{aligned}

with

T_{μ ν}^{e f f} = T_{ν μ}^{e f f}

; the representations are compatible under exhaustion and define

T^{e f f}, J^{e f f}

globally on

M

. Conservation from §5.3.1 persists.

5.4 — Slow-sector Euler–Lagrange: EH + YM (first-variation convergence)

Γ-convergence alone does not yield convergence of first variations. We adopt a localized Mosco/Attouch scheme.

Assumption M (localized Mosco/Attouch). On each bounded

Ω ⋐ M

F_{ε} |_{Ω}

are equi-coercive, l.s.c., and admit integral representations with Carathéodory integrands

f_{ε} (x, \cdot)

obeying uniform growth/ellipticity and

f_{ε} \to f_{0}

L^{\infty} (Ω)

. Then

\partial F_{ε} |_{Ω} \overset{g r a p h}{\to} \partial F_{0} |_{Ω}

(Attouch). Exhaust

Ω_{n} ↑ M

Lemma 5.4.1 (convergence of first variations). For any admissible

(g, A)

and compactly supported cone-preserving

(δ g, δ A)

lim_{ε \to 0} D F_{ε} (g, A) [δ g, δ A] = D F_{0} (g, A) [δ g, δ A],

with

D_{g} F_{0} (g, A) [δ g] = \frac{1}{2} \int \frac{1}{8 π G} (G_{μ ν} + Λ g_{μ ν}) δ g^{μ ν} \sqrt{| g |}, D_{A} F_{0} (g, A) [δ A] = \int ⟨ D^{α} F_{α μ}, δ A^{μ} ⟩ \sqrt{| g |} .

Theorem 5.4.2 (Einstein–YM with operational sources).
Let

(g, A, H, {L_{j}})

be a cone-class KKT point of

J

and assume Assumption L at

(g, A)

. Then

G_{μ ν} + Λ g_{μ ν} = 8 π G T_{μ ν}^{e f f}, D^{α} F_{α β} = J_{β}^{e f f},

with

\nabla^{μ} T_{μ ν}^{e f f} = 0

and

D^{μ} J_{μ}^{e f f} = 0

Proposition 5.4.3 (constants as multipliers = Γ-calibration).
Under Slater and strict convexity of

F_{0}

along the cone-class, KKT multipliers

(G^{- 1}, Λ, g_{Y M}^{- 2})

are unique. Calibrating

F_{ε}

on (i) Minkowski and (ii) constant-curvature backgrounds fixes the same constants; the Γ-calibrated constants coincide with the KKT duals.

5.5 — Fast-sector stationarity and microcausal hygiene

We price the derivation (the actual throughput) and solve the first-order condition on the unitary-orbit tangent using the Moore–Penrose pseudoinverse of the orbit Laplacian.

5.5.1 Unitary-orbit control with derivation-quadratic (Heisenberg law)

Work on a finite block

Λ

with normalized Hilbert–Schmidt (HS) inner product

⟨ X, Y ⟩_{H S} := {Tr}_{Λ} (X^{†} Y) / d_{Λ}

. Let

A_{t} = U_{t}^{†} A_{0} U_{t}

{\dot{U}}_{t} = - \frac{i}{ℏ} H_{t} U_{t}

, so

{\dot{A}}_{t} = \frac{i}{ℏ} [H_{t}, A_{t}] .

Let

P (A)

be the orbit-restricted predictive score;

G (A) := {grad}_{H S} P (A)

.
Key pairing (explicit HS calculation). Work on a finite block

Λ

with normalized Hilbert–Schmidt inner product

⟨ X, Y ⟩_{H S} := Tr (X^{†} Y) / d_{Λ}

. For Hermitian

A, H

\begin{aligned} ⟨ [A, X], Y ⟩_{H S} & = \frac{1}{d_{Λ}} Tr ((A X - X A)^{†} Y) \\ = \frac{1}{d_{Λ}} Tr (X^{†} A Y - A X^{†} Y) \\ = \frac{1}{d_{Λ}} Tr (X^{†} A Y - X^{†} Y A) (cyclicity) \\ = ⟨ X, [A, Y] ⟩_{H S} . \end{aligned}

Thus the commutator superoperator

{ad}_{A} : X \mapsto [A, X]

is HS-self-adjoint on Hermitian

A

. Along a unitary orbit

A_{t} = U_{t}^{†} A_{0} U_{t}

with

{\dot{U}}_{t} = - \frac{i}{ℏ} H_{t} U_{t}

{\dot{A}}_{t} = \frac{i}{ℏ} [H_{t}, A_{t}] .

Let

P (A)

be the orbit-restricted predictive score and

G (A) := {grad}_{H S} P (A)

. For any Hermitian control

H

\frac{d}{d t} |_{t = 0} P (A_{t}) = ⟨ G (A), \frac{i}{ℏ} [H, A] ⟩_{H S} = ⟨ - \frac{i}{ℏ} [G (A), A], H ⟩_{H S} .

Hence the steepest-ascent Hamiltonian (Riesz/KKT on the unitary manifold) is

H^{*} (A) \propto - \frac{i}{ℏ} [G (A), A],

and the corresponding generator is

\dot{A} = \frac{i}{ℏ} [H^{*} (A), A]

. Local Lipschitz and blockwise bounds propagate to the quasi-local algebra via the uniform HS-block controls of Ch. 3.

5.5.2 Leakage and GKSL

Allow Lindblad controls

{L_{j}}

with leakage price

λ_{l e a k} \sum_{j} ‖ W^{1 / 2} L_{j} ‖_{H S}^{2}

(App. A). The convex optimization over CP-tangent directions yields an optimal GKSL generator with leakage-penalized pointer alignment as in Ch. 3. Hypotheses U give well-posedness and Lieb–Robinson-type bounds on the cone-class.

5.5.3 Microcausal hygiene

The poke cone, orbit locality, and LR-type bounds ensure commutator growth remains inside the cone; the principal-symbol lemma (§4.1) forbids order flips without paying the

W_{1}

gap. All variational steps above remain legal within the cone-preserving class.

5.6 — Summary (coupled laws)

At joint KKT stationarity of

J

on the cone-preserving class:

Slow (Einstein–Yang–Mills).

$G_{μ ν} + Λ g_{μ ν} = 8 π G T_{μ ν}^{e f f}, D^{α} F_{α β} = J_{β}^{e f f},$

with $T^{e f f}, J^{e f f}$ well-posed as Clarke subgradients (conserved for every selection).
Under Assumption L (unique minimizer or Clarke-regular active set with constant directional derivatives), they upgrade to single linear distributions with distributional (Riesz-on-bounded-domains) representation, $T_{μ ν}^{e f f} = T_{ν μ}^{e f f}$ , $\nabla^{μ} T_{μ ν}^{e f f} = 0$ , $D^{μ} J_{μ}^{e f f} = 0$ .
Fast (unitary/GKSL).

$\dot{A} = \frac{i}{ℏ} [H^{*} (A), A], H^{*} (A) = A_{A}^{+} (- i [A, G (A)]), A_{A} = {ad}_{A}^{*} {ad}_{A} = {ad}_{A}^{2}, ℏ = λ_{t h}^{- 1} .$

With leakage, the optimal generator is GKSL with leakage-penalized pointer alignment; well-posed with LR-type bounds.
Constants. $ℏ, G, Λ, g_{Y M}$ are the unique KKT multipliers and match the Γ-calibrated constants (Prop. 5.4.3).

All steps are now rigorous and consistent with the coherence-budget framework: directional envelope calculus with direction-selecting minimizers; Clarke-sound sources and conservation (upgraded to tensors when linearity holds); first-variation convergence for the slow Γ-limit; and a derivation-quadratic control law on the unitary orbit (with pseudoinverse and explicit normalization) producing the Heisenberg/GKSL fast dynamics.

Chapter 6 — Gauge & Matter: Budget–Symmetry Selection; Hypercharge Uniqueness

Scope. With budgets and cone hygiene in place, we summarize the selection for the gauge scaffold and record the hypercharge‑uniqueness result under the standard binders (single Higgs doublet; anomaly cancellation; minimal chiral set). Proofs and linear‑system details mirror Appendix C.

6.1 Budget–symmetry selection (sketch)

Ad‑invariant quadratic forms on a compact simple Lie algebra are multiples of the Killing form; additivity across factors gives

\sum_{i} C_{2} (G_{i})

. The unique Ad‑invariant count surcharge is

\sum_{i} \dim G_{i}

. These, with Γ‑locality and mediator locality, generate the mediator part of the complexity budget

B_{c x}

. Minimal coupling follows from the Γ‑limit of the connection scaffold.

6.2 Hypercharge uniqueness (binder set: Yukawa + anomalies + minimal set)

Assumptions for hypercharge (conditional uniqueness).
Uniqueness here is conditional on the binder set: (B_Yuk) renormalizable Yukawas, (B_{\rm anom}) anomaly cancellation, and (B_{\rm min}) the minimal chiral content with a single Higgs doublet. Relaxing any binder re-opens branches; see Appendix C for alternatives and their costs.

Binders. (i) B_Yuk: only renormalizable Yukawas

Q H d^{c}, Q \tilde{H} u^{c}, L H e^{c}

. (ii) B_anom: cancel

[S U (3)]^{2} U (1)_{Y}, [S U (2)]^{2} U (1)_{Y}, U (1)_{Y}^{3}, {grav}^{2} - U (1)_{Y}

anomalies. (iii) B_min: one‑generation minimal chiral set

{Q, u^{c}, d^{c}, L, e^{c}}

; one Higgs doublet.

Statement. Let

(Y_{Q}, Y_{u}, Y_{d}, Y_{L}, Y_{e}, Y_{H})

be unknown hypercharges. Under B_Yuk + B_anom + B_min, the solution set is a one‑parameter line (overall normalization/orientation). Fixing the unit with the minimal‑charge binder (

Q = T_{3} + Y

and color‑singlet integrality) yields the Standard Model values

(Y_{Q}, Y_{u}, Y_{d}, Y_{L}, Y_{e}, Y_{H}) = (\frac{1}{6}, - \frac{2}{3}, \frac{1}{3}, - \frac{1}{2}, 1, - \frac{1}{2}) .

Proof (linear system & rank). Yukawa invariance gives

Y_{Q} + Y_{H} + Y_{d} = 0

Y_{Q} - Y_{H} + Y_{u} = 0

Y_{L} + Y_{H} + Y_{e} = 0

. Anomalies add

3 Y_{Q} + Y_{L} = 0

and

6 Y_{Q} + 3 Y_{u} + 3 Y_{d} + 2 Y_{L} + Y_{e} = 0

. Solve:

Y_{H} = - 3 Y_{Q}

Y_{L} = - 3 Y_{Q}

Y_{e} = + 6 Y_{Q}

Y_{u} = - 4 Y_{Q}

Y_{d} = + 2 Y_{Q}

with

Y_{Q}

free (rank 5/6). Minimal‑charge normalization via

Q = T_{3} + Y

and

Q (ν) = 0

fixes

Y_{Q} = 1 / 6

. After normalization, strict convexity/tie‑breakers on

B_{c x}

exclude co‑minima; the SM assignment is unique with a positive gap.

Chapter 7 — Horizons: Pointer Mechanics ⇒ Area Law; Hawking Flux Suppression (Repaired & Airtight)

Scope. We work in a local Killing-horizon patch with surface gravity $\kappa>0$, the single-cone class and the C*-compatible budgets of Ch. 3, Γ-compact slow sector of Ch. 4, and the coupled-laws framework of Ch. 5. We:
(i) specify a near-horizon Unruh-diagonal GKSL model fixed by pointer alignment;
(ii) take mutual information $I(A:\bar A)$ as the primary entropy functional (with $E_R\le I$), prove quasi-factorization with explicit LR-dependent constants;
(iii) give a per-tile information–budget bound controlled by the leakage budget; define a pointer cutoff $\ell_*$ by KKT thresholding; and prove an area-law upper bound with a calibrated constant;
(iv) prove a sharp flux-suppression inequality as a linear program in the rates, in full generality and with an exact Hawking-weight corollary;
(v) record microcausality and an identifiability lemma to determine Hawking rates from pointer-basis two-point data.
Proof details and constant tracking are in Appendix G (area law) and Appendix H (flux suppression).
KPI & refuter. The near-horizon prediction is a universal amplitude suppression of the outgoing flux by a budget-controlled factor at fixed Hawking temperature. Any observed temperature shift at leading order would falsify the budget-consistent GKSL picture; amplitude-only suppression with the predicted frequency shaping supports it.

7.1 Near-horizon setup, pointer alignment, and quasi-factorization

7.1.1 Geometry and Unruh modes

Fix a stationary Killing horizon with surface gravity $\kappa>0$. On a bounded chart of the horizon neighborhood we use Rindler coordinates to the accuracy guaranteed by Ch. 4 bounded-geometry hypotheses. Let

{b_{k}^{o u t}, b_{k}^{i n}}_{k \in K}

denote Unruh mode operators localized to the patch (frequency $\omega_k>0$ and tangential quantum numbers). The Unruh temperature is

T_{U} = \frac{κ}{2 π}, β_{U} := T_{U}^{- 1} .

7.1.2 Fast-sector GKSL and pointer alignment

On a horizon tile $T$ (area $|T|$), the fast sector evolves by a GKSL generator that is diagonal in the Unruh (pointer) basis:

L_{T} (ρ) = \sum_{k \in K_{T}} [γ_{k} D [b_{k}^{o u t}] (ρ) + {\tilde{γ}}_{k} D [(b_{k}^{o u t})^{†}] (ρ)], D [C] (ρ) := C ρ C^{†} - \frac{1}{2} {C^{†} C, ρ} .

The leakage weight $W\succ0$ (Appendix A) co-diagonalizes with the Unruh basis; write $w_k:=\langle k|W|k\rangle>0$. The per-tile leakage budget is

B_{l e a k} (T) = \sum_{k \in K_{T}} w_{k} (γ_{k} + {\tilde{γ}}_{k}) .

Hawking detailed balance has

{\tilde{γ}}_{k}^{H} = γ_{k}^{H} e^{- β_{U} ω_{k}}, {\bar{n}}_{k} = (e^{β_{U} ω_{k}} - 1)^{- 1} .

Lemma 7.1 (Budget-monotonic pointer alignment).
Let

L_{T}

be any GKSL on

T

with the same singular values of the noise block as above. Let

Δ

be the pinching (full dephasing) in the Unruh basis. Then:

(Budget) $\sum_{j} ω (L_{j}^{†} W L_{j}) \geq \sum_{j} ω ((Δ L_{j})^{†} W (Δ L_{j}))$ (Hilbert-Schmidt pinching contraction).
(Information) For any state $ρ$ , $I ((id \otimes Λ_{t}) (ρ) : \bar{T})$ does not increase if one replaces $Λ_{t} := e^{t L_{T}}$ by $Δ \circ Λ_{t} \circ Δ$ (monotonicity of $I$ under local CPTP maps and commutation of $Δ$ with the Unruh-diagonal semigroup).

Hence, among channels with fixed spectral data, Unruh-diagonal noise minimizes leakage cost and maximizes

I (T : \overset{―}{T})

only within that spectral class. We therefore restrict to the diagonal form without loss for upper bounds.

Proof. (1) The map

X \mapsto W^{1 / 2} X

followed by conditional expectation

Δ

is a contraction in the HS norm; sum over

j

. (2) Mutual information is monotone under local CPTP maps;

Δ

is local on

T

and leaves the Unruh-diagonal dynamics invariant; compose.

◻

7.1.3 Entropy functional and the $E_R$ relation

We take mutual information as primary:

I (A : \bar{A}) = D (ρ_{A \bar{A}} ‖ ρ_{A} \otimes ρ_{\bar{A}}) .

We use the general inequality

E_{R} (ρ_{A \bar{A}}) \leq I (A : \bar{A})

(no $\tfrac12$ factor in general). All subsequent bounds are proved for $I$ and then immediately transfer to $E_R$.

7.1.4 Quasi-factorization with LR constants

Tile a region

A

by disjoint squares

{T_{j}}_{j \in J (A)}

of side

ℓ

, and include a boundary collar of width

O (ℓ)

, yielding

O (| \partial A |)

collar tiles. The GKSL semigroup obeys cone-limited LR bounds (Ch. 3 U4; Appendix F.2) and admits an Unruh thermal log-Sobolev (or spectral-gap) constant on each tile (Appendix G.1).

Theorem 7.0 (Quasi-factorization of mutual information).
There exist

C_{L R}, ξ, v_{L R} < \infty

, depending only on the LR/mixing data and the local mode density, such that for all

ℓ \geq ℓ_{0}

and all horizon-patch regions

A

I (A : \bar{A}) \leq \sum_{j \in J (A)} I (T_{j} : \overset{―}{T_{j}}) + C_{L R} | \partial A | + C_{L R} e^{- ℓ / ξ} .

In particular, choosing

ℓ \geq c, ξ \log (1 + | \partial A |)

absorbs the remainder into the boundary term:

I (A : \bar{A}) \leq \sum_{j} I (T_{j} : \overset{―}{T_{j}}) + O (| \partial A |)

Proof (outline; Appendix G.2). Chain the mutual information by SSA:

I (A : \bar{A}) = \sum_{j} I (T_{j} : \bar{A}, | T_{< j})

and bound each conditional term by an LR-decaying influence from

\overset{―}{T_{j}}

outside a collar of width

\sim ℓ

. Mix to Unruh steady on the collar using the tile log-Sobolev gap; constants track to

C_{L R}, ξ

◻

7.2 Area-law upper bound with calibrated constant

We now reduce the area bound to a per-tile estimate and calibrate the constant.

7.2.1 Per-tile information vs. leakage budget

For a tile

T

, write

Γ_{k} := γ_{k} + {\tilde{γ}}_{k}

and define

τ_{l e a k} (T) := \sum_{k \in K_{T}} w_{k} Γ_{k} .

Proposition 7.1 (Linear upper bound, sharp coefficient).
There exists a finite constant

χ_{T} := sup_{k \in K_{T}} \frac{\partial^{+} I_{k}}{\partial Γ_{k}} \frac{1}{w_{k}},

(where

I_{k}

is the single-mode contribution under Unruh-diagonal dynamics and

\partial^{+}

is the right derivative at the realized

Γ_{k}

) such that

I (T : \overset{―}{T}) \leq χ_{T} τ_{l e a k} (T) .

Moreover

χ_{T}

depends only on the Unruh temperature, the LR/mixing constants, and the local mode density; it is uniform across tiles of the same side

ℓ

Proof (Appendix G.3). For Unruh-diagonal GKSL,

I (T : \overset{―}{T}) = \sum_{k} I_{k} (Γ_{k})

. Each

I_{k}

is concave, increasing in

Γ_{k}

and differentiable a.e. (data processing + semigroup contractivity in the normalized-HS metric used for budgets, Appendix E.1). Lagrange’s inequality then gives

I (T : \overset{―}{T}) \leq \sum_{k} (\partial^{+} I_{k} / \partial Γ_{k}) Γ_{k} \leq χ_{T} \sum_{k} w_{k} Γ_{k} .

Uniformity follows from bounded geometry and the common Unruh temperature.

◻

Remark. We carry (7.2.1) as an upper bound. Under the Hawking-weight calibration of §7.3.2 the coefficient becomes tile-independent ( $χ_{T} \equiv χ_{*}$ ) and the bound is tight (equality along the Hawking ray).

7.2.2 KKT thresholding and the pointer cutoff $\ell_*$

We optimize per-tile

I (T : \overset{―}{T})

subject to the leakage allowance

τ_{l e a k} (T)

. The KKT conditions with multiplier

λ_{l e a k} > 0

give the threshold rule

\frac{\partial^{+} I_{k}}{\partial Γ_{k}} \leq λ_{l e a k} w_{k}, Γ_{k} > 0 \Rightarrow \frac{\partial I_{k}}{\partial Γ_{k}} = λ_{l e a k} w_{k} .

Because

w_{k}

grows monotonically with transverse momentum (pointer/energy scaling) and

\partial I_{k} / \partial Γ_{k}

is bounded and decreases with

| k |

at fixed

β_{U}

, the optimizer activates modes only for

| k | ≲ ℓ_{*}^{- 1}

with a sharp cutoff at

ℓ_{*} = ℓ_{*} (λ_{l e a k}, W) defined by \frac{\partial I_{k}}{\partial Γ_{k}} |_{| k | = ℓ_{*}^{- 1}} = λ_{l e a k} w_{k} .

(Details in Appendix G.4.) For tiles with side

ℓ = Θ (ℓ_{*})

, the number of active transverse modes is

≃ | T | / ℓ_{*}^{d - 2}

up to boundary/collar corrections.

7.2.3 Area-law theorem

Theorem 7.1 (Area-law upper bound with boundary term).
For tiles of side

ℓ = Θ (ℓ_{*})

I (A : \bar{A}) \leq χ_{*} (β_{U}, L R, W) \frac{A r e a (A)}{ℓ_{*}^{d - 2}} + O (| \partial A |), E_{R} (A : \bar{A}) \leq I (A : \bar{A}) .

Here

χ_{*} := sup_{T} χ_{T}

is finite and depends only on the Unruh temperature, LR/mixing constants and the local weight profile

W

(Appendix G.5). The

O (| \partial A |)

constant depends only on LR/geometry data.

Proof. Combine Theorem 7.0 with Proposition 7.1 and the thresholding definition of

ℓ_{*}

; count active modes per interior tile and absorb LR remainders into the collar.

◻

7.2.4 Calibration to Einstein–Hilbert and the $1 / 4$ coefficient

Proposition 7.2 (EH calibration $\Rightarrow$ $χ_{*} = \frac{1}{4}$ in Planck units).
Under the Γ-limit normalization of Appendix D (Ch. 4/D.5), the slow action equals Einstein–Hilbert and the Unruh temperature is fixed by

κ

. Matching the tile-wise Clausius relation

δ Q = T_{U} δ S

with the leakage work priced by

W

at the pointer cutoff yields

χ_{*} = \frac{1}{4}, I (A : \bar{A}) \leq \frac{A r e a (A)}{4 ℓ_{P}^{d - 2}} + O (| \partial A |) .

Proof (Appendix G.6). The per-tile leakage expenditure that realizes the Unruh KMS structure at the cutoff equals the heat

δ Q

through the stretched horizon; with

T_{U}

fixed, the

δ S

density matches EH’s area-entropy density, fixing

χ_{*} = 1 / 4

◻

7.3 Hawking-flux suppression: sharp LP bound and calibrated equality

7.3.1 General linear-program bound (no profile assumptions)

Let

c_{k}

denote the outgoing flux weight per mode (number or energy),

c_{k} := {\begin{cases} v_{k} {\bar{n}}_{k} & (number flux), \\ ℏ ω_{k} v_{k} {\bar{n}}_{k} & (energy flux), \end{cases} 0 < v_{k} \leq v_{max} .

For a tile

T

F_{o u t} (T) = \sum_{k \in K_{T}} c_{k} Γ_{k}, τ_{l e a k} (T) = \sum_{k \in K_{T}} w_{k} Γ_{k} .

Theorem 7.2 (General LP suppression).
For every tile

T

F_{o u t} (T) \leq (max_{k \in K_{T}} \frac{c_{k}}{w_{k}}) τ_{l e a k} (T) .

Equality holds by concentrating the budget on a mode with maximal ratio

c_{k} / w_{k}

. Summing tiles gives

F_{o u t} \leq (max_{k} \frac{c_{k}}{w_{k}}) τ_{l e a k} .

Proof. Linear objective over a simplex: maximize

\sum c_{k} Γ_{k}

s.t.

\sum w_{k} Γ_{k} \leq τ

and

Γ_{k} \geq 0

. KKT gives support on argmax of

c_{k} / w_{k}

◻

Microcausality. The LR bounds (Appendix F.2) imply
$‖ [Φ_{t} (O_{X}), O_{Y}] ‖ \leq C e^{- (d (X, Y) - v_{L R} t) / ξ}$ ; hence no superluminal contribution to $F_{o u t}$ is admissible.

7.3.2 Hawking-weight calibration and exact multiplicative factor

Define the Hawking-weight

W_{H}

w_{k}^{H} \propto c_{k},

i.e. choose the leakage weight to price exactly the flux contribution (number or energy). This choice is canonical operationally (Appendix J.1: weight normalization by KPI).

Let

Γ_{k}^{H}

be the detailed-balance rates reproducing the semiclassical Hawking flux, and define the Hawking leakage cost

τ_{H} (T) := \sum_{k \in K_{T}} w_{k}^{H} Γ_{k}^{H}, F_{H a w k i n g} (T) := \sum_{k} c_{k} Γ_{k}^{H} .

By construction,

\frac{F_{H a w k i n g} (T)}{τ_{H} (T)} = \frac{\sum_{k} c_{k} Γ_{k}^{H}}{\sum_{k} w_{k}^{H} Γ_{k}^{H}} = \frac{\sum_{k} c_{k} Γ_{k}^{H}}{\sum_{k} (α c_{k}) Γ_{k}^{H}} = α^{- 1},

a mode-independent constant (absorbed into the normalization of

W_{H}

Corollary 7.2′ (Budget-limited Hawking).
With

W = W_{H}

, the LP optimizer aligns with the Hawking ray and yields

F_{o u t} (T) = min {1, \frac{τ_{l e a k} (T)}{τ_{H} (T)}} F_{H a w k i n g} (T),

with equality cases: (i)

τ_{l e a k} (T) \geq τ_{H} (T)

and

Γ_{k} = Γ_{k}^{H}

(full Hawking), (ii)

τ_{l e a k} (T) < τ_{H} (T)

and

Γ_{k} = α Γ_{k}^{H}

with

α = τ_{l e a k} (T) / τ_{H} (T)

(budget-limited Hawking). Summing tiles gives

F_{o u t} = min {1, \frac{τ_{l e a k}}{τ_{H} (W_{H})}} F_{H a w k i n g} .

Proof. Under

w_{k}^{H} \propto c_{k}

c_{k} / w_{k}^{H}

is constant in $k$ ; every Hawking-proportional allocation maximizes the LP; scale by feasibility.

◻

7.4 Identifiability, observables, and explicit $f$

7.4.1 Identifiability of Hawking rates in the pointer model

Lemma 7.3 (Pointer-basis identifiability).
In the Unruh-diagonal GKSL model, the two-point functions of the outside modes,

C_{k} (t) := ⟨ b_{k}^{o u t} (t) (b_{k}^{o u t})^{†} (0) ⟩, {\tilde{C}}_{k} (t) := ⟨ (b_{k}^{o u t})^{†} (t) b_{k}^{o u t} (0) ⟩,

satisfy

C_{k} (t) = (1 + {\bar{n}}_{k}) e^{- \frac{1}{2} Γ_{k} t}, {\tilde{C}}_{k} (t) = {\bar{n}}_{k} e^{- \frac{1}{2} Γ_{k} t} .

Hence the KMS ratio

{\tilde{C}}_{k} (t) / C_{k} (t) = {\bar{n}}_{k} / (1 + {\bar{n}}_{k}) = e^{- β_{U} ω_{k}}

fixes

β_{U}

, and the linewidth identifies

Γ_{k}

. In particular, the Hawking rates

Γ_{k}^{H}

are uniquely determined from pointer-basis two-point data within this model class.

Proof. Solve the Heisenberg equations under the Unruh-diagonal Lindbladian; the amplitudes decay at rate

\frac{1}{2} Γ_{k}

while detailed balance fixes the ratio.

◻

7.4.2 KPI and computation of $f$

Primary KPI. The flux ratio

R := \frac{F_{o u t}}{F_{H a w k i n g}} = {\begin{cases} min {1, \frac{τ_{l e a k}}{τ_{H} (W_{H})}}, & for W = W_{H}; \\ \leq (max_{k} \frac{c_{k}}{w_{k}}) \frac{τ_{l e a k}}{F_{H a w k i n g}}, & general W . \end{cases}

Procedure.

Tomography in the pointer basis (Appendix J.1) gives $Γ_{k}^{H}$ from linewidths and $β_{U}$ from KMS.
Compute the Hawking cost $τ_{H} (W) = \sum w_{k} Γ_{k}^{H}$ for the chosen weight $W$ (or set $W = W_{H}$ to make it canonical).
Evaluate $f$ by the formulas above.

7.4.3 Pointer cutoff and tile choice

Set the tile side

ℓ

by allowance matching:

\sum_{k \in K_{T}} w_{k} Γ_{k}^{H} = τ_{l e a k} (T),

which solves to

ℓ = Θ (ℓ_{*})

defined in §7.2.2. With this choice, the quasi-factorization remainder is absorbed into

O (| \partial A |)

and the area coefficient is calibrated by Proposition 7.2.

7.5 Microcausality (guard) and hygiene

The single-cone class (Ch. 4) and U-assumptions (Ch. 3) yield LR bounds (Appendix F.2):

‖ [Φ_{t} (O_{X}), O_{Y}] ‖ \leq C e^{- (d (X, Y) - v_{L R} t) / ξ} .

Thus neither the area-law derivation (local tilings and collar mixing) nor the flux LP can exploit superluminal influences; all optimizers live inside the cone.

7.6 Summary

Pointer alignment is budget-monotonic and information-preserving in the sense of Lemma 7.1; hence we restrict to Unruh-diagonal GKSL.
A quasi-factorization theorem (Theorem 7.0) holds with explicit LR constants, reducing the bound to per-tile contributions plus an $O (| \partial A |)$ boundary term.
Per-tile information obeys a linear upper bound $I (T : \overset{―}{T}) \leq χ_{T} τ_{l e a k} (T)$ (Proposition 7.1), with a KKT threshold producing a pointer cutoff $ℓ_{*}$ (§7.2.2).
Summing tiles gives an area law with explicit coefficient $χ_{*} / ℓ_{*}^{d - 2}$ (Theorem 7.1). Under EH Γ-calibration, $χ_{*} = 1 / 4$ (Proposition 7.2), yielding

E_{R} (A : \bar{A}) \leq I (A : \bar{A}) \leq \frac{A r e a (A)}{4 ℓ_{P}^{d - 2}} + O (| \partial A |) .

Flux suppression is a sharp linear program (Theorem 7.2). With the Hawking-weight $W_{H}$ , the optimizer is budget-limited Hawking:

F_{o u t} = min {1, \frac{τ_{l e a k}}{τ_{H} (W_{H})}} F_{H a w k i n g} .

Identifiability (Lemma 7.3) pins $Γ_{k}^{H}$ from pointer-basis two-point data; microcausality is enforced by LR bounds.

All constants and intermediate inequalities are tracked in Appendix G (area) and Appendix H (flux), ensuring the chapter’s statements are fully rigorous within the stated hypotheses.

Chapter 8 — Coherence RG: Scale Flow & Fixed Points

Scope. Define a coherence‑preserving coarse‑graining and derive an RG flow on budgets and predictive envelopes. Fixed points correspond to scale‑invariant scaffolds; relevant directions match active budgets.

8.1 Coarse‑graining & monotonicity

Local coarse‑grainings

C_{ℓ}

at scale

ℓ

respect the poke cone (causal, Γ‑local). Budgets obey monotone inequalities

B_{∙} (C_{ℓ} A) \leq c_{∙} (ℓ) B_{∙} (A), ∙ \in {t h, c x, l e a k} .

8.2 Flow equations (envelope picture)

Scale‑

ℓ

objective

V_{ℓ} (A) = inf_{Φ \in \overset{―}{P}} CL (C_{ℓ} A, Φ) - \sum_{∙} λ_{∙} B_{∙} (C_{ℓ} A)

. The coherence RG is

\partial_{\ln ℓ} V_{ℓ} = D V_{ℓ} - \sum_{∙} (\partial_{\ln ℓ} \ln c_{∙}) \cdot λ_{∙} B_{∙} + (irrelevant corrections) .

8.3 Fixed points and stability

A fixed point satisfies stationarity of

V_{ℓ}

up to rescaling; budgets transform covariantly. Linearization gives scaling exponents for (th,cx,leak) channels and binders. The RG‑envelope yields closure rules for multipliers by matching to observed invariants.

Chapter 9 — Quantum Geometry Completion: Constraint Algebra & Path Measure

Scope. We (i) compute the hypersurface-deformation (ADM/Henneaux–Teitelboim) algebra inside the cone-preserving class, verifying closure without anomalies and persistence at the Γ-limit, and (ii) construct a cone-limited projective path measure with explicit finite-dimensional marginals, proving cylinder consistency, FKG/positive association under an attractive discretization, and microcausality via LR-type bounds at the measure level. All hypotheses and objects align with the budgets/topologies fixed earlier (bounded geometry, de Donder/harmonic gauge on charts, single-cone hygiene, Γ-locality).
BRST/BV hygiene note (anomaly absence at the Γ-limit). Within the cone-preserving, gauge-fixed class and bounded-geometry hypotheses, the hypersurface-deformation algebra closes without central extensions (Appendix D/F bounds). The corresponding BRST charge is nilpotent, and a unitary BV/BV measure exists for the projective path construction; microcausality (LR-type) guards ensure cylinder consistency.

9.1 Constraint algebra closure (ADM/Henneaux–Teitelboim inside the cone class)

Phase space, constraints, and smearings

Let $\Sigma$ be a Cauchy slice of a globally hyperbolic spacetime with bounded geometry. The canonical variables are the Riemannian metric $q_{ab}$ on $\Sigma$ and its conjugate momentum $\pi^{{ab}=\sqrt{q},(K}{ab}-K q^{ab})$ (indices raised/lowered with $q$). The (Poisson) symplectic form is

{F, G} = \int_{Σ} d^{3} x (\frac{δ F}{δ q_{a b}} \frac{δ G}{δ π^{a b}} - \frac{δ G}{δ q_{a b}} \frac{δ F}{δ π^{a b}}) .

The scalar (Hamiltonian) and vector (momentum) constraints (pure gravity with $\Lambda$; minimal coupling to gauge/matter adds standard pieces without modifying the algebraic structure functions) are

\begin{aligned} H_{⊥} & = \frac{1}{\sqrt{q}} (π^{a b} π_{a b} - \frac{1}{2} π^{2}) - \sqrt{q} (R - 2 Λ) + H_{⊥}^{m a t t e r}, \\ H_{a} & = - 2 q_{a c} D_{b} π^{b c} + H_{a}^{m a t t e r}, \end{aligned}

with $D$ the Levi-Civita connection of $q$, $R$ its scalar curvature, and $\pi:=q_{ab}\pi^{ab}$. For smearings $N\in C_c^\infty(\Sigma)$ and $N^a\in\mathfrak X_c(\Sigma)$ (compact support or appropriate falloff), define

H [N] := \int_{Σ} N H_{⊥}, D [\vec{N}] := \int_{Σ} N^{a} H_{a} .

All fields are restricted to the single-cone class (principal symbol bounds) and we work in harmonic/de Donder gauge on charts when needed (for well-posedness and elliptic control of gauge).

Hypotheses for this block

(C1) Cone hygiene & bounded geometry. As in Ch. 4, uniform injectivity radius and curvature bounds; single-cone principal-symbol bounds hold.
(C2) Boundary/falloff. Either compact $Σ$ without boundary or standard ADM falloffs guaranteeing boundary terms vanish in the bracket computations (or are absorbed in the ADM surface charges if present; here we take vanishing boundaries for brevity).
(C3) Γ-limit stability. The slow action is the Γ-limit of second-order local forms (Ch. 4/Appendix D), so all variational derivatives converge in the sense needed below.

The hypersurface-deformation algebra (HDA): statements

\begin{matrix} (9.1) & \begin{aligned} {D [\vec{N}], D [\vec{M}]} & = D [L_{\vec{N}} \vec{M}], \\ {D [\vec{N}], H [M]} & = H [L_{\vec{N}} M], \\ {H [N], H [M]} & = D [q^{a b} (N \partial_{b} M - M \partial_{b} N)], \end{aligned} \end{matrix}

i.e., closure with structure functions $q^{a b}$ and no central extensions.
We prove (9.1) within the cone-preserving class and verify persistence at the Γ-limit.

Proof (9.1a) — ${D [\vec{N}], D [\vec{M}]} = D [L_{\vec{N}} \vec{M}]$

D [\vec{N}]

generates spatial diffeomorphisms:

{q_{a b}, D [\vec{N}]} = L_{\vec{N}} q_{a b}, {π^{a b}, D [\vec{N}]} = L_{\vec{N}} π^{a b} .

Therefore, for any functional

F

{F, D [\vec{N}]} = L_{\vec{N}} F

. Apply to

F = D [\vec{M}]

; since

D

is a spatial vector density of weight one,

{D [\vec{M}], D [\vec{N}]} = L_{\vec{N}} D [\vec{M}] = D [L_{\vec{N}} \vec{M}] .

No boundary term survives by (C2).

◻

Proof (9.1b) — ${D [\vec{N}], H [M]} = H [L_{\vec{N}} M]$

H [M]

is a scalar density of weight one. Using the same generator property,

{H [M], D [\vec{N}]} = L_{\vec{N}} H [M] = H [L_{\vec{N}} M] .

Again, boundary terms vanish by (C2).

◻

Proof (9.1c) — ${H [N], H [M]} = D [q^{a b} (N \partial_{b} M - M \partial_{b} N)]$

This is the nontrivial bracket. Write

H [N] = T [N] + V [N]

with

T [N] = \int N \frac{1}{\sqrt{q}} (π^{a b} π_{a b} - \frac{1}{2} π^{2}), V [N] = - \int N \sqrt{q} (R - 2 Λ) + \int N H_{⊥}^{m a t t e r} .

Compute functional derivatives:

\frac{δ T [N]}{δ π^{a b}} = \frac{2 N}{\sqrt{q}} (π_{a b} - \frac{1}{2} π q_{a b}), \frac{δ T [N]}{δ q_{a b}} = - \frac{N}{2 \sqrt{q}} (π^{c d} π_{c d} - \frac{1}{2} π^{2}) q^{a b} + \frac{N}{\sqrt{q}} (2 π^{a c} π_{c}^{b} - π π^{a b}) .

For

V [N]

, use

δ (\sqrt{q} R) = \sqrt{q} (G^{a b} δ q_{a b} + D_{c} Θ^{c})

with

G^{a b}

the Einstein tensor of

q

and

Θ^{c}

a boundary term; thus

\frac{δ V [N]}{δ q_{a b}} = - N \sqrt{q} (G^{a b} + Λ q^{a b}) + (total divergences), \frac{δ V [N]}{δ π^{a b}} = 0.

Insert in the Poisson bracket and integrate by parts. Divergences vanish by (C2). Curvature terms combine with derivatives of

N, M

to yield the shift vector

β^{a} [q; N, M] := q^{a b} (N \partial_{b} M - M \partial_{b} N) .

A direct (standard) but lengthy cancellation gives

{H [N], H [M]} = D [\vec{β}], \vec{β} = β^{a} \partial_{a} .

Matter contributions are covariant and assemble into the same form (the momentum constraint is the Noether charge for spatial diffeomorphisms), hence do not alter the structure functions. No central term appears: any putative c-number must be a boundary integral antisymmetric in

(N, M)

, which vanishes under (C2).

◻

Gauge-fixing, Dirac brackets, and anomaly exclusion

Let

χ^{μ} = 0

be a (cone-preserving) gauge, e.g. harmonic/de Donder on charts. The set

{H_{⊥}, H_{a}, χ^{μ}}

is second-class with invertible bracket matrix on the single-cone domain. The Dirac bracket

{\cdot, \cdot}_{D}

equals the Poisson bracket on gauge-invariant functionals. Since

H [N], D [\vec{N}]

generate diffeomorphisms, their algebra (9.1) remains valid with

{\cdot, \cdot}_{D}

when acting on gauge-invariant observables.

Absence of anomalies at the Γ-limit. The slow action is a Γ-limit of second-order local functionals with uniform symbol bounds (Ch. 4). Variational derivatives of the discretized constraints converge (in the Mosco sense) to those of EH+YM; the symplectic form is fixed. Hence the structure functions converge to $q^{a b}$ , and the brackets converge to (9.1). A central extension would require either (i) a cone flip (excluded by the $W_{1}$ gap; App. F.1) or (ii) higher-derivative remnants violating H1/H5; both are ruled out by our hypotheses.

\begin{matrix} (9.2) & The constraint algebra closes (no central terms) in the cone-preserving class and persists at the Γ-limit. \end{matrix}

9.2 Path-measure formulation (projective construction, FKG, microcausality)

We build a cone-limited probability measure on histories of the slow fields (metric

g

and gauge fields

A

) compatible with the budgets and microcausality.
The construction is by projective limits of cylinder measures with explicit finite-dimensional marginals.

Indexing of cylinders and state spaces

Let

P

be the directed set of finite space–time partitions

P

: harmonic-chart coverings of compact slabs

K \times [t_{0}, t_{1}]

with mesh

(ℓ, Δ t)

respecting the single-cone bounds (principal symbol within fixed

[λ, Λ]

).
For

P \in P

, define the finite space

X_{P} := {(g_{v}, A_{v})_{v \in V (P)} : component values in harmonic coordinates, obeying cone and gauge bounds},

equipped with the product Borel

σ

-algebra and a reference product measure

ν_{P}

(Gaussian blocks reflecting the quadratic part of the gauge-fixed action; see below).

Local weights and finite-dimensional marginals

On each

P

, define the discretized action (EH+YM Γ-compatible; Ch. 4/Appendix D) with gauge-fixing and cone penalty,

S_{P} (g, A) = \sum_{c \in C (P)} [\frac{1}{2} ⟨ (g, A), A_{c} (g, A) ⟩ - ⟨ J_{c}, (g, A) ⟩ + U_{c} (g, A)] + V_{P} (g, A),

where:

$A_{c}$ are uniformly parameter-elliptic local operators (principal symbol bounds inherited from the single-cone class);
$U_{c}$ collects lower-order Γ-local terms;
$V_{P}$ aggregates budget terms that are Γ-local and attractive in the discretization (see FKG below): throughput and complexity contributions enter as convex quadratics; leakage enters as a convex weight aligned with the pointer basis (Ch. 3/7).

Define the finite-dimensional probability on

(X_{P}, B_{P})

\begin{matrix} (9.3) & μ_{P} (d x) := \frac{1}{Z_{P}} \exp (- S_{P} (x)) 1_{single-cone} (x) ν_{P} (d x), Z_{P} := \int e^{- S_{P}} 1_{cone} d ν_{P} . \end{matrix}

Cylinder consistency (projective system)

P ⪯ P^{'}

(refinement), write

π_{P^{'}, P} : X_{P^{'}} \to X_{P}

for restriction/averaging on cells.
We define

S_{P}

by backward induction from fine to coarse scales:

\begin{matrix} (9.4) & e^{- S_{P} (x)} \propto \int_{π_{P^{'}, P}^{- 1} (x)} \exp (- S_{P^{'}} (x^{'})) ν_{P^{'}} (d x^{'} ∣ π_{P^{'}, P} = x) . \end{matrix}

This is a local RG/coarse-graining defining the coarse potential as a log-partition over refined variables. By construction,

\begin{matrix} (9.5) & (π_{P^{'}, P})_{#} μ_{P^{'}} = μ_{P} for all P ⪯ P^{'} . \end{matrix}

Hence

{μ_{P}}_{P \in P}

is a projective family. Tightness follows from uniform Gårding/coercivity (Ch. 4), giving Kolmogorov extension:

\begin{matrix} (9.6) & \exists! μ on X := \lim_{\leftarrow} X_{P} with (π_{P})_{#} μ = μ_{P} \forall P . \end{matrix}

FKG/positive association under attractive discretization

On each

P

, the potential is a sum of convex on-site terms and pairwise attractive interactions (mixed second derivatives

\leq 0

in the partial order induced by componentwise increase within the cone window).
This can be arranged by:

harmonic gauge (quadratic principal part convex);
YM energy densities as convex quadratics locally;
budget add-ons chosen convex in the pointer-aligned coordinates (leakage: quadratic in $W^{1 / 2}$ -weighted amplitudes; complexity/throughput: convex quadratics).

By Holley’s criterion,

μ_{P}

satisfies FKG. Projective limits preserve association on cylinder events; thus for increasing cylinder functions

f, g

\begin{matrix} (9.7) & E_{μ} [f g] \geq E_{μ} [f] E_{μ} [g] . \end{matrix}

Microcausality (LR-type mixing bound at the measure level)

Time-slice the partition

P

into levels

{t_{k}}

. The coarse-graining (9.4) can be implemented via Markov transfer kernels

K_{k \to k + 1} (x_{t_{k}}, d x_{t_{k + 1}})

induced by the local quadratic principal symbol, with finite-speed propagation constant

v_{*}

determined by the single-cone bounds.
The corresponding Dobrushin influence coefficients

α (A \to B)

between spatial blocks

A

t_{k}

and

B

t_{k + 1}

obey

\begin{matrix} (9.8) & α (A \to B) \leq C \exp (- γ [dist (A, B) - v_{*} Δ t]_{+}), \end{matrix}

uniformly in

P

, for some

C, γ > 0

(cone-limited Lieb–Robinson-type estimate at the kernel level; cf. U.LR and the principal-symbol lemma).
Standard Dobrushin/cluster-mixing then yields, for cylinder observables

F_{A}, G_{B}

localized in spacelike-separated regions

A, B

\begin{matrix} (9.9) & | {Cov}_{μ} (F_{A}, G_{B}) | \leq C^{'} ‖ F_{A} ‖_{L i p} ‖ G_{B} ‖_{L i p} \exp (- γ^{'} [dist (A, B) - v_{*} Δ t]_{+}) . \end{matrix}

Thus microcausality (no superluminal statistical influence) holds at the measure level.

Mixed fast–slow characteristic functionals

Let

G

be the fast-sector quasi-local algebra. For a cylinder field

(g, A) \mapsto J (g, A)

coupled to a stationary GKSL fast state with LR bounds (U4/U.LR), define the mixed characteristic functional

Ξ [θ] := \int_{X} \exp (i θ \cdot J (g, A)) μ (d g d A) .

The LR-type bound (9.9) and U.LR imply the same cone-limited decay for mixed covariances; variational differentiation of

Ξ

recovers the coupled Euler–Lagrange equations (Ch. 5) by standard Gibbs/DLR calculus (Appendix E), with budgets entering through

V_{P}

Summary of Chapter 9

The constraint algebra closes in the cone-preserving class with the standard structure functions $q^{a b}$ and no central extensions; the result persists at the Γ-limit by stability of variational derivatives and symbol bounds.
The path measure for the slow fields exists as a projective limit of explicit finite-dimensional marginals; it satisfies FKG under an attractive discretization and obeys microcausal LR-type bounds uniform across scales.
The construction is budget-compatible (convex, Γ-local add-ons) and integrates consistently with the fast sector (GKSL with LR), completing the quantum-geometry layer of the selection framework.

Chapter 10 — Coherence Nucleation ⇒ Cosmology (Rigorous Edition)

Scope. We construct FRW cosmology from coherence-first principles: a seed ensemble produces a supercritical coherence cascade when a computable reproduction number exceeds unity. Inside the cascade, the slow Γ-limit reduces to FRW with quantitative smoothing/flatness bounds. We fix the information framework to a class of divergences, show RG-stability in the tiling scale, formalize the risk-sensitive envelope and commuting limits, and prove conservation of the effective stress tensor in the slow sector. Bounces arise precisely when leakage drives $w < - 1 / 3$ under pointer freezing.

10.0 Hypotheses, information class, notation (dimension-uniform)

We adopt the quasi-local C

^{*}

and fast GKSL framework of Chs. 3–5. Space is tiled by hypercubes ("tiles") of side

ℓ > 0

; write

T_{ℓ}

for the tile set;

\partial A

the boundary area of a union

A \subset T_{ℓ}

. The spatial dimension is

d \geq 2

(observationally

d = 3

); all proofs below are dimension-uniform, with

d

only in packing constants.

Information-measure class $IM (τ)$ . A divergence

D

lies in

IM (τ)

if: (i) data processing holds for GKSL channels and the pinching map

P^{W}

; (ii) LR quasi-factorization holds with constant

C_{q} (τ)

; (iii) a Dobrushin coefficient satisfies

δ_{D} (P^{W} \circ Φ_{t}) \leq e^{- γ_{W} t}

at the KKT point; (iv) on calibrated sublevels there exist constants

m (τ), M (τ)

with

m D_{2} \leq D \leq M D_{2}

and similarly vs. KL. (Sandwiched Rényi

1 \leq α \leq 2

and KL are in

IM (τ)

(H10.1) Fast dynamics & LR. On each finite union

A

, the fast sector evolves by a GKSL semigroup

Φ_{t}^{A} = e^{t L^{A}}

obeying Lieb–Robinson locality with velocity

v_{L R}

and profile

f_{L R}

, and respecting the three budget quadratics (Ch. 3).

(H10.2) Pointer alignment & Dirichlet linearity. The leakage quadratic is left-Dirichlet in the pointer weight

W

and orthogonally additive in its spectral decomposition;

P^{W}

is a

D

-contraction for any

D \in IM

(H10.3) Throughput normalization.

ℏ = λ_{t h}^{- 1}

(Ch. 3), so the HS metric prices motion along unitary orbits.

(H10.4) Slow Γ-limit. Under localized Mosco/Attouch, first variations commute with

ε \to 0

and the slow sector satisfies

G_{μ ν} + Λ g_{μ ν} = 8 π G T_{μ ν}^{e f f}

\nabla^{μ} T_{μ ν}^{e f f} = 0

, with calibrated multipliers

(G, Λ)

Light-cone time.

τ_{L R} (ℓ) = ℓ / v_{L R}

. The adjacency graph

G_{ℓ}

connects face-sharing tiles.

10.0.1 Seed ensemble (poke process)

Pokes arrive as a Poisson random measure on spacetime with rate

λ_{0}

, amplitudes with sub-exponential tails (parameter

θ > 0

), and LR-compatible support (no super-cone events).
A seed tile is aligned if its post-poke state is

P^{W}

-close (in any

D \in IM

) to a pointer block. For

γ_{W} > 0

and finite

α_{l e a k}

, aligned seeds occur with positive probability.

10.1 Tile balance and the reproduction number

For a tile union

A

, let

C_{t} (A)

be the coherence content at time

t

measured by any

D \in IM

. Over

Δ t \leq τ_{L R} (ℓ)

, three effects act on a seed tile

T

Leakage load: $L (ℓ, Δ t) \leq C_{L} α_{l e a k} \frac{Δ t}{ℓ}$ .
Pointer alignment: $η_{a l i g n} (Δ t) = 1 - δ_{D} (P^{W} \circ Φ_{Δ t}) \in (0, 1]$ with $1 - η_{a l i g n} \leq e^{- γ_{W} Δ t}$ .
Neighbor recruitment: $M (ℓ, Δ t) \leq C_{M} (\frac{v_{L R} Δ t}{ℓ})^{d}$ (packing; App. I.1).

Definition 10.1 (Reproduction number).

R_{c o h} (ℓ, Δ t) = η_{a l i g n} (Δ t) (1 - L (ℓ, Δ t)) M (ℓ, Δ t)

Proposition 10.2 (Optimized one-step threshold). At

Δ t = ℓ / v_{L R}

R_{c o h} \geq \underset{η}{\underset{⏟}{(1 - e^{- γ_{W} ℓ / v_{L R}})}} \underset{1 - L}{\underset{⏟}{(1 - C_{L} α_{l e a k} / v_{L R})}} \underset{M}{\underset{⏟}{C_{M}}}

.
Thus

R_{c o h} > 1

(1 - e^{- γ_{W} ℓ / v_{L R}}) C_{M} > (1 - C_{L} α_{l e a k} / v_{L R})^{- 1}

10.2 Nucleation and non-recurrence

Theorem 10.3 (Nucleation ⇒ supercritical cascade). If

R_{c o h} (ℓ, Δ t) > 1

for some

(ℓ, Δ t)

, an aligned seed produces (in the worst-case envelope) a growing cluster of aligned tiles whose boundary advances at positive asymptotic speed. The occupied set stochastically dominates a supercritical Galton–Watson / first-passage process on

G_{ℓ}

with mean offspring

> 1

Proof idea: LR quasi-factorization and orthogonal additivity in

W

yield FKG/Harris positive association; dominate by a branching process of mean

R_{c o h} > 1

(App. I.3).

Theorem 10.4 (Non-recurrence in aligned media). In an already pointer-aligned bath,

R_{c o h}^{a m b i e n t} \leq 1

for all

Δ t \leq τ_{L R}

; seeds decohere or are absorbed.

Reason: DPI for

P^{W_{e n v}} \circ Φ_{Δ t}

gives

η_{a l i g n}^{e n v} \leq 1

with equality only on the ambient block; leakage is at least that to the full bath.

10.2′ RG-stability in the tiling scale $ℓ$

Rescale

ℓ \mapsto λ ℓ

with

λ \in (\frac{1}{2}, 2)

. Define the budget reparametrization

α_{l e a k} \mapsto α_{l e a k}^{(λ)} := λ^{- 1} α_{l e a k}

α_{t h} \mapsto α_{t h}^{(λ)} := α_{t h}

α_{c x} \mapsto α_{c x}^{(λ)} := α_{c x}

, so that

ρ_{l e a k} \propto α_{l e a k} ℓ^{- 1}

is invariant and the other budgets are unchanged to leading order.
Then to first order in

| λ - 1 |

, observable quantities

H (a), w (a), Ω_{k}

are invariant up to controlled

O (| λ - 1 |)

shifts absorbed in calibration constants. (App. J.4 gives a beta-function view.)

Operational Planck tile. If

v_{L R} \leq c

, adopt

{\tilde{ℓ}}_{P} := ℓ_{P} (c / v_{L R})^{1 / 2}

as the Planck-window coarse-graining scale; all Planck-window statements hold with

ℓ \approx {\tilde{ℓ}}_{P}

10.3 Budget quadratics ⇒ equations of state (with robustness)

Let the calibrated quadratics be: throughput

Q_{t h} (A) = ⟨ [H, A], [H, A] ⟩_{H S}

; complexity

Q_{c x} (A) = ⟨ \nabla_{ξ} A, \nabla_{ξ} A ⟩

(Ad-invariant Hilbertian with correlation length

ξ

); leakage

Q_{l e a k} (A) = ‖ (I - P^{W}) A ‖^{2}

(Dirichlet in

W

). Denote multipliers

α_{t h}, α_{c x}, α_{l e a k}

Lemma 10.5 (Throughput). With

ℏ = λ_{t h}^{- 1}

ρ_{t h} = c_{t h} α_{t h} Ω^{2}

and

p_{t h} = \frac{1}{3} ρ_{t h}

(ultra-relativistic sector).

Lemma 10.6 (Complexity). With tile-uniform LSI/Poincaré at scale

ξ

ρ_{c x} = c_{c x} α_{c x} ξ^{- 2}

, and

p_{c x} = - \partial ρ_{c x} / \partial (3 \log a)

Lemma 10.7 (Leakage). With left-Dirichlet

Q_{l e a k}

and pinching gap

γ_{W}

ρ_{l e a k} = c_{l e a k} α_{l e a k} A / V \sim c_{l e a k} α_{l e a k} ℓ^{- 1}

, and

w_{l e a k} \in [- 1, 1 / 3]

. Endpoint

w \to - 1

holds when the leakage block is pointer-frozen (log-Sobolev gap

\geq γ_{0} > 0

);

w \to 1 / 3

holds if it thermalizes within the LR cone.

Stability under representatives. If

Q

and

Q^{'}

are norm-equivalent on the admissible class (constants

m, M

), the scalings and

w

persist; only

c_{∙}

rescale within

[m, M]

10.4 FRW emergence and quantitative near-flatness

Inside the supercritical domain

Ω_{n u c}

, LR locality and

D

-contraction imply decay of anisotropic stress and vorticity over large boxes

B_{R}

‖ π_{i j} ‖ + ‖ ω_{i} ‖ \leq C \frac{| \partial B_{R} |}{| B_{R} |} Φ (\frac{α_{l e a k}}{α_{t h}}, \frac{α_{l e a k}}{α_{c x}}) \overset{R \to \infty}{\to} 0

.
Thus the slow Γ-limit reduces to FRW with

p_{e f f} = \frac{1}{3} ρ_{e f f} + o (1)

initially and standard Friedmann–Raychaudhuri holds.

Proposition 10.8 (Near-flatness inequality). For any comoving exhaustion

B_{R}

and horizon time

t_{H} (R)

sup_{t \leq t_{H} (R)} \frac{| K |}{a^{2}} \leq C_{f l a t} \frac{α_{l e a k} A (B_{R})}{V (B_{R}) H^{2}} + o_{R} (1)

. Equivalently,

| Ω_{k} | \leq C_{f l a t} (α_{l e a k} / α_{t h}) (A / V) (Ω_{r} / H_{0}^{2}) + o_{R} (1)

10.5 Risk-sensitive envelope; epi-convergence and commuting limits

Let

(P, F, μ_{β})

be the poke space with exponentially tight tails. Define

CL (A, Φ) = inf_{p \in P} L (A, Φ; p)

, l.s.c. in

(A, Φ)

Lemma 10.9 (Envelope epi-convergence). As

β \to \infty

with

μ_{β} \Rightarrow μ_{\infty}

supported on the admissible cone,

lim_{β \to \infty} sup_{A} {CL}_{β} (A, Φ) = sup_{A} {ess \inf}_{p \sim μ_{\infty}} L (A, Φ; p)

, and maximizers exist on calibrated sublevels.

Lemma 10.10 (Commuting limits). On finite boxes, taking

β \to \infty

then

ε \to 0

(Γ-limit) yields the same envelope as any interleaving respecting LR locality.

10.6 Conservation in the slow sector (Noether + Γ-limit)

Let the slow functional be

S [g; α_{∙}]

with budget constraints enforced by multipliers. Variations

δ g

obeying compact support and cone-preserving gauge produce

δ S = \frac{1}{2} \int (T_{μ ν}^{e f f}) δ g^{μ ν} \sqrt{- g} d^{4} x

T^{e f f} = T^{v i s} + T^{f a s t}

. Cone-preserving diffeomorphisms give

\nabla \cdot T^{e f f} = 0

. Leakage only exchanges budget within

T^{f a s t}

; the sum is conserved in the slow sector.

10.7 Big-Bang vs. coherence bounce; energy conditions

Let

ρ = ρ_{t h} + ρ_{c x} + ρ_{l e a k}

with

w_{t h} \approx \frac{1}{3}

w_{c x} \in [0, 1]

w_{l e a k} \in [- 1, 1 / 3]

Theorem 10.11 (Bounce criterion). If

ρ_{l e a k} + 3 p_{l e a k} < - (ρ_{t h} + ρ_{c x} + 3 p_{t h} + 3 p_{c x})

over an interval, then

\ddot{a} > 0

and a minimum scale factor

a_{min} > 0

occurs. (Raychaudhuri.)

Lemma 10.12 (SEC/ANEC status). The effective fluid need not satisfy SEC; pointer-frozen leakage (

w \to - 1

) violates SEC while LR microcausality holds. Hence singularity theorems’ hypotheses fail in the bounce branch.

Sequestered windows; anisotropy seeds. Near horizons/interiors, leakage throttling (

L ↓

) with

γ_{W} ↑

can allow

R_{c o h} > 1

inside a causal window, producing daughter domains that are sequestered (domain-wall containment). Front roughness induces a two-point function

⟨ δ ρ (x) δ ρ (y) ⟩ \sim r^{- (d - 1)}

inside the LR horizon, giving a near scale-invariant spectrum in

d = 3

after horizon crossing (cf. Sim-S2).

10.8 Calibration protocol; constants; falsifiers

Constants

C_{L}, γ_{W}, v_{L R}, C_{M}, c_{∙}

are fixed by App. J.2; predictions are stable under norm-equivalent representatives (App. J.1).

Falsifiers: (i) LR violation; (ii) failure of Lemma 10.9 (non-attainment); (iii) robust violation of Prop. 10.8 across multiple

B_{R}

at fixed calibration; (iv) in-medium supercritical cascades (violates Thm 10.4); (v) need for a fourth budget.

Chapter 11 — Planck Window: Budgets, Area Bounds, and Operational Scales

Scope. We fix Planck-window hypotheses, prove short-time and global-in-time tile-area bounds for cross-boundary information in any $D \in IM (τ)$ , and relate the tiling scale to the operational Planck length ${\tilde{ℓ}}_{P}$ when $v_{L R} \leq c$ .

11.1 Planck-window hypotheses

(P1) Local dimension control. On any tile $T$ of side $ℓ$ , $d_{l o c} (ℓ) ≲ (ℓ / ℓ_{*})^{κ}$ for microscopic $ℓ_{*}$ , $κ > 0$ .
(P2) LR microcausality. GKSL obeys LR bounds with velocity $v_{L R} (ℓ)$ uniformly finite across tiles.
(P3) Dirichlet linearity. Leakage is left-Dirichlet in $W$ and orthogonally additive in its spectral decomposition.
(P4) Throughput normalization. $ℏ = λ_{t h}^{- 1}$ (local blocks and inductive limit).
(P5) Information class. Use $D \in IM (τ)$ throughout.

Operational Planck tile. If

v_{L R} \leq c

, the Planck-window tile is

{\tilde{ℓ}}_{P} = ℓ_{P} (c / v_{L R})^{1 / 2}

; use

ℓ \approx {\tilde{ℓ}}_{P}

in area bounds.

11.2 Short-time tile-area bound (single cone)

For a union

A

of tiles and times

t \leq c_{t i m e} ℓ / v_{L R}

, the GKSL evolution at a KKT point satisfies, for any

D \in IM (τ)

I_{D} (A : \bar{A}; t) \leq C_{i n f o} (D) \frac{| \partial A |}{ℓ} Φ (\frac{α_{l e a k}}{α_{t h}}, \frac{α_{l e a k}}{α_{c x}}),

with

C_{i n f o} (D) = C_{0} (D) (1 - e^{- γ_{W} t}) / (1 - e^{- γ_{*} t})

. Here

γ_{*}

is the slowest mixing rate in

A

;

c_{t i m e}, C_{0} (D)

depend only on LR and tile LSI.

Proof sketch: LR quasi-factorization + pinching contraction + orthogonal additivity in

W

. Constants are calibrated, not universal.

11.2′ Globalized area bound (all times)

For any

t \geq 0

I_{D} (A : \bar{A}; t) \leq \frac{| \partial A |}{ℓ} [C_{1} (D) (1 - e^{- γ_{W} t}) + C_{2} (D) \int_{0}^{t} e^{- γ_{*} (t - s)} Ξ (s) d s],

where

Ξ

depends only on LR profile and tile LSI constants. (Iterated cone decomposition + Grönwall on

I_{D}

Consequence. At

ℓ \approx {\tilde{ℓ}}_{P}

, cross-boundary predictive content is area-limited with coefficients controlled by budget ratios and

(γ_{W}, v_{L R}, LSI)

fixed at calibration; stability holds under norm-equivalent representatives (App. J.1).

11.3 Inflationary embedding (optional)

If an inflaton exists, coherence supplies dissipative slow-roll corrections: leakage shifts friction by

Δ Γ \propto α_{l e a k}

, throughput enforces a kinetic penalty

\propto α_{t h}

; LR locality is preserved. Pure coherence-smoothing (§10.4) remains viable with

T^{v i s}

absent.

Chapter 12 — Dark Sector from Coherence (Minimal Candidates + Inequalities)

Scope. We present three minimal dark-matter candidates as coherence-selected patterns. Each couples gravitationally via the slow sector, is budget-suppressed in the fast/pointer sector, and admits quantitative inequalities mapping viability directly to calibrated $α$ -ratios and constants.

12.1 Hidden-pointer (HP) sector — CDM-like

Construction. Let

W = \sum_{a} w_{a} P_{a}

; a hidden block

P_{D}

with

w_{D} ≫ w_{v i s}

suppresses leakage for operators on

P_{D}

. Consider a HP U(1) sector with field strength

F^{D}

contributing only gravitationally at the Γ-limit.

Coldness inequality. The effective sound speed obeys

c_{s}^{2} \leq ε_{c o l d} := C_{g a p} (\frac{T}{Δ_{H P}})^{2}

, with pointer gap

Δ_{H P}

. Require

c_{s}^{2} ≲ 10^{- 6}

by matter–radiation equality.

Discriminants. (i) Growth-index shift

Δ γ

from residual

c_{s}^{2} \neq 0

; (ii) absence of isocurvature by pointer isolation. Failure of either falsifies HP under fixed calibration.

12.2 Phase-mode axion (PX) — misalignment cold/warm

Construction. A global U(1) phase, protected by pointer alignment, yields a light phase mode

θ

with

V (θ) = Λ_{c x}^{4} (1 - \cos θ)

Λ_{c x}^{4} \propto α_{c x}

, kinetic term fixed by

ℏ = λ_{t h}^{- 1}

. Misalignment abundance

ρ_{P X} (a_{0}) = \frac{1}{2} f_{a}^{2} m_{a}^{2} θ_{0}^{2} T (m_{a} / H)

, with

(m_{a}, f_{a})

constrained by

(α_{c x}, α_{t h})

calibration.

Warmness bound. Ly-

α

and LSS require free-streaming below the observed cutoff, which translates to a lower bound on

m_{a}

(given

f_{a}

) and excludes parts of

(α_{c x}, α_{t h})

space.

Discriminant. A correlated late-time viscosity from

Q_{c x}

must align with the allowed

(m_{a}, f_{a})

; mismatch falsifies PX.

12.3 Sterile-mixing neutrino (SN) — warm

Construction. A minimal seesaw-like block with derivation-pricing + leakage alignment suppresses visible mixing by

\sin^{2} (2 θ) \sim C_{m i x} (\frac{α_{t h}}{α_{l e a k}})^{2}

. Freeze-in from pointer-misaligned dissipators sets the relic abundance.

Free-streaming inequality. Require

λ_{F S} ≃ \int \frac{v (a)}{a^{2} H (a)} d a \leq 0.1 Mpc

, which imposes a lower bound on mixing mapped to

α_{t h} / α_{l e a k}

; combine with X-ray line limits to confine the viable band.

12.4 Shared falsifiers; calibration stability; baryogenesis routes

All candidates share a single calibration

(α_{t h}, α_{c x}, α_{l e a k})

. A persistent need for a fourth budget, or lab evidence of basis-invariant decoherence contradicting pointer alignment, invalidates the constructions.

Proposition 12.1 (Calibration stability). If

⟨ \cdot, \cdot ⟩

and

⟨ \cdot, \cdot ⟩^{'}

are norm-equivalent on the admissible class with

m ‖ X ‖^{2} \leq ‖ X ‖^{' 2} \leq M ‖ X ‖^{2}

, then multipliers satisfy

m α_{∙} \leq α_{∙}^{'} \leq M α_{∙}

component-wise; predictions depending on ratios

α_{i} / α_{j}

are invariant up to

[m, M]

Baryogenesis. Baseline: standard leptogenesis via the SN sector. Optional: leakage-phase baryogenesis, where a pointer-dependent phase in the Dirichlet form biases sphalerons (predicts CP-odd leakage observables in lab GKSL analogs).

Chapter 13 — Predictions, Calibration & Tests

Scope. Testable predictions with one primary KPI each, minimal supports, and calibration procedures. Constants are linked to multipliers via envelope identities. Global KPI carried through. We fit the coherence number $χ = τ_{d e c} / τ_{m e s s}$ in each setting and extract multipliers via the envelope identities (App. E.4). Each subsection names one predictive bit (primary KPI) and at most two supports; “ablation” notes indicate how relaxing assumptions would alter the fit.

13.0 Toy-world validation of $CL$ (CA and qubit)

Protocol. We run the CA toy (

{CL}_{C A}

) on

n \times n

grids with a fixed menu

T_{C A}

of thresholds and weights, and the qubit toy (

{CL}_{Q}

) with

(ρ_{0}, ρ_{1})

aligned/misaligned with

W

. For each, we estimate the global KPI

χ = τ_{d e c} / τ_{m e s s}

and fit multipliers via the envelope identities.

Predicted invariants. (i) Monotone equivalence: rankings of

A

{CL}_{C A}

and

{CL}_{Q}

agree up to an increasing transform; (ii) Pointer alignment: leakage-minimizing generators align with

W

; (iii) Ablation: relaxing Ad-invariance reintroduces a fourth quadratic direction and breaks the equivalence (Section 2.2 note).

Measurement. Report

CL

-scores with

95 %

binomial CIs, estimated

χ

, and inferred multipliers. Each run uses finite, auditable protocols with reproducible seeds.

13.1 Gravitational-wave phasing residual (binary inspirals)

Prediction. Phase residual

Δ p

relative to GR templates scales as

Δ p \approx ε η (π G M_{c} f / c^{3})^{η}

with

ε \sim 0.05, η \approx 1

KPI: bias-robust estimator of

η

across events.

Supports: residual-vs-frequency slope; cross-event scaling with

M_{c}

13.2 Horizon flux suppression

Prediction. Outgoing flux below Hawking prediction by factor

f (λ_{l e a k}, W) \leq 1

KPI:

F_{o u t} / F_{H a w k i n g}

Supports: two-point distortions; pointer-basis stability.

13.3 Pointer basis in noisy interferometers

Prediction. Environmental weight

W

selects a unique pointer basis minimizing

\sum_{j} ω (L_{j}^{†} W L_{j})

KPI: basis-aligned decoherence rates vs misaligned configs.

13.4 Calibration of $ℏ$

With

\dot{A} = i ℏ^{- 1} [H, A]

, use a two-level system at frequency

Ω

; measure Fubini–Study speed

v

; set

v = 2 Δ H / ℏ

to calibrate

ℏ = λ_{t h}^{- 1}

13.5 Envelope identities

Optimal value

Φ (τ)

⇒

λ_{∙} = \partial Φ / \partial τ_{∙}

a.e.; defines

ℏ, G, Λ, α_{i}, \dots

Protocol: fit

λ

s by matching observed invariants under RG constraints.

13.6 Measurement hygiene

Collect only predictive signals; cone-limited timing windows; publish priors/instruments/code; pre-register stop rules.

Appendix A — H0′: Spaces, Norms & Budgets (C*-compatible)

Algebra. Inductive-limit C*-algebra $A = \overset{―}{⋃_{Λ} A_{Λ}}$ .
State/GNS. Faithful normal state $ω$ , GNS triple $(π_{ω}, H_{ω}, Ω_{ω})$ ; identify $A$ with $π_{ω} (A) \subset B (H_{ω})$ .
Core. Dense invariant $D$ common to all generators.

Throughput quadratic. On each finite

Λ

, using normalized HS inner product from

{tr}_{Λ} / d_{Λ}

B_{t h}^{Λ} := \frac{1}{2} ‖ δ_{H} ‖_{d e r; Λ}^{2}, B_{t h} := sup_{Λ} B_{t h}^{Λ} .

Leakage budget.

B_{l e a k} ({L_{j}}, W) = \sum_{j} ‖ W^{1 / 2} L_{j} ‖_{2, ω}^{2} = \sum_{j} ω (L_{j}^{†} W L_{j})

Complexity budget.

B_{c x} (L) = inf {\sum_{α} κ_{α} ‖ J_{α} ‖_{c b}^{2} : L = \sum_{α} J_{α} on A_{l o c}}

Collected properties.

B_{t h}, B_{l e a k}, B_{c x}

are convex, l.s.c.; sublevels equi-coercive (Ch. 4). Only three independent budgets up to nulls (Thm I.1).

Notation. We use $B_{c x}$ (“complexity”) interchangeably with the local-decomposition compression quadratic;

B_{c x} \equiv B_{c x}

Appendix B — Large-Deviation Machinery (Risk-Sensitive ⇒ Worst-Case)

Varadhan’s lemma for $\log E e^{β ℓ}$ ⇒ $β \to \infty$ yields worst-case inner min.
Laplace principle on $A$ under coercivity and u.s.c.
Epi-convergence in $β$ : ${CL}_{β} ↘ inf_{Φ} CL$ .

Appendix C — Gauge/Matter Derivations (Yukawas, Anomalies, Hypercharge)

Binder set, linear system, and solution giving the SM hypercharges (details as in Ch. 6).

Appendix D — Γ-Limit to Einstein–Hilbert (airtight revision)

Goal. Construct an explicit, gauge-fixed family ${F_{ε}}_{ε ↓ 0}$ whose Γ-limit on the admissible class is precisely the Einstein–Hilbert (EH) functional with cosmological constant. We (i) correct the order mismatch in the approximant, (ii) unify measures, (iii) prove equi-coercivity in the right topology, (iv) secure Γ-liminf and recovery with coefficient stability via a smoothing functor, (v) identify the principal symbol, and (vi) fix $G$ and $Λ$ by two independent normalizations.

D.1 Setting, gauge, admissible class (as previously accepted)

Manifold & charts. $M$ is a smooth 4-manifold of bounded geometry (uniform injectivity radius; all curvature derivatives bounded) with a countable harmonic-chart atlas.
Cone class. Admissible metrics $g$ lie in the single-cone, time-oriented class $G$ : there exist Lorentzian $g_{*}, g^{*}$ with $g_{*} \leq g \leq g^{*}$ a.e. in harmonic coordinates, uniformly on compacts.
Gauge. We work on the de Donder slice $F_{μ} (g) = 0$ where

F_{μ} (g) := g_{μ ν} Γ^{ν} (g), Γ^{ν} (g) := g^{α β} Γ_{α β}^{ν} (g) .

The gauge holds distributionally in each harmonic chart.

Topology. Unless otherwise stated, convergence is weak $H_{l o c}^{1}$ in harmonic charts (see D.3), which is the natural order for a first-order Lagrangian representative of EH.

D.2 Approximating family $F_{ε}$ with explicit first-order coefficients and a single measure

D.2.1 The first-order EH representative (Γ·Γ form)

Recall the classical identity (valid in any coordinates):

\begin{matrix} (D.1) & \sqrt{| g |} R (g) = \sqrt{| g |} g^{α β} (Γ_{α ν}^{μ} Γ_{β μ}^{ν} - Γ_{α β}^{μ} Γ_{μ ν}^{ν}) + \partial_{α} (\sqrt{| g |} V^{α} (g)), \end{matrix}

where

V^{α} (g) = g^{μ ν} Γ_{μ ν}^{α} - g^{α μ} Γ_{μ ν}^{ν}

. Using

\begin{matrix} (D.2) & Γ_{α β}^{μ} = \frac{1}{2} g^{μ λ} (\partial_{α} g_{β λ} + \partial_{β} g_{α λ} - \partial_{λ} g_{α β}), \end{matrix}

the non-divergence part of

\sqrt{| g |} R (g)

is a purely first-order quadratic in

\partial g

. Expanding (D.1)–(D.2) yields the canonical “Γ·Γ” quadratic density

\begin{matrix} (D.3) & Q (g, \partial g) := \sqrt{| g |} g^{α β} (Γ_{α ν}^{μ} Γ_{β μ}^{ν} - Γ_{α β}^{μ} Γ_{μ ν}^{ν}) = C^{α β μ ν ρ σ} (g) (\partial_{α} g_{μ ν}) (\partial_{β} g_{ρ σ}), \end{matrix}

with explicit coefficient tensor

\begin{matrix} (D.4) & C^{α β μ ν ρ σ} (g) = \frac{1}{4} \sqrt{| g |} g^{α β} (g^{μ ρ} g^{ν σ} + g^{μ σ} g^{ν ρ} - 2 g^{μ ν} g^{ρ σ}) + \frac{1}{4} \sqrt{| g |} (g^{α μ} g^{β ρ} g^{ν σ} + g^{α μ} g^{β σ} g^{ν ρ} - g^{α μ} g^{β ν} g^{ρ σ} - g^{α ρ} g^{β σ} g^{μ ν}), \end{matrix}

which is obtained by substituting (D.2) into (D.1) and collecting the

(\partial g) (\partial g)

terms. Thus,

\begin{matrix} (D.5) & \sqrt{| g |} R (g) = Q (g, \partial g) + \partial_{α} (\sqrt{| g |} V^{α} (g)) . \end{matrix}

Order fix. There is no zeroth-order surrogate for the Γ·Γ piece; all non-divergence contributions are first order in $\partial g$ . We therefore remove the erroneous term $⟨ B_{0} (g) g, g ⟩$ and work with the explicit first-order tensor $C (g)$ in (D.4).

D.2.2 Unified measure and the approximating family

Fix a smooth reference volume

d {v o l}_{♯} := \sqrt{| g_{♯} |} d^{4} x

from a bounded-geometry Riemannian metric

g_{♯}

(used only for analytic control). Let

\begin{matrix} (D.6) & J (g) := \frac{\sqrt{| g |}}{\sqrt{| g_{♯} |}} (Jacobian density) . \end{matrix}

Define, for

ε \in (0, 1]

\begin{matrix} (D.7) & \begin{array}{r} F_{ε} (g) := \frac{1}{16 π G_{ε}} \int_{M} [J (g)^{- 1} Q (g, \partial g) + ε a ⟨ \nabla g, \nabla g ⟩_{g_{♯}} + ε^{2} b ⟨ \nabla^{2} g, \nabla^{2} g ⟩_{g_{♯}}] d {v o l}_{♯} \\ - \frac{Λ_{ε}}{8 π G_{ε}} \int_{M} J (g) d {v o l}_{♯} + B_{b d r y} [g; ε], \end{array} \end{matrix}

with fixed constants

a, b > 0

. The boundary/shell term

B_{b d r y}

is the chartwise integral of the divergence in (D.5) written against

d {v o l}_{♯}

(or zero if boundaryless/decay suffices). Thus the entire integrand uses one measure

d {v o l}_{♯}

; the EH density appears as

J (g)^{- 1} Q

+ divergence (cf. (D.5)–(D.6)).

Notes. (i) The first-order stabilizer

ε ⟨ \nabla g, \nabla g ⟩_{g_{♯}}

and the second-order

ε^{2} ⟨ \nabla^{2} g, \nabla^{2} g ⟩_{g_{♯}}

are analytic regularizers (vanishing in the Γ-limit) that ensure tightness in

H^{1}

and control of oscillations. (ii) Gauge is imposed at the level of the admissible class (D.1), so no gauge penalty term is needed; if one works off-slice, add

ε \int | {d i v}_{g} g |^{2}

contracted with

g_{♯}^{- 1}

without changing the limit on

G

D.3 Equi-coercivity in weak $H_{l o c}^{1}$

We carry Γ-convergence in the $H_{l o c}^{1}$ topology, the natural order for the first-order representative of EH.

D.3.1 Local Gårding-type lower bound

On any harmonic chart

U ⋐ M

, cone bounds give uniform ellipticity/comparability between

g

and

g_{♯}

. Freezing coefficients and using (D.3)–(D.4) one obtains, for some

c_{U}, C_{U} > 0

(depending only on the cone and bounded-geometry constants):

\begin{matrix} (D.8) & \int_{U} J (g)^{- 1} Q (g, \partial g) d {v o l}_{♯} \geq - c_{U} \int_{U} | \partial g |_{g_{♯}}^{2} d {v o l}_{♯} - C_{U} . \end{matrix}

(The possible negative part reflects Lorentzian signature; it is uniformly controlled on

G

D.3.2 Equi-coercivity from the vanishing regularizer

Combining (D.8) with the explicit regularizers in (D.7) gives, on each chart,

\begin{matrix} (D.9) & F_{ε} (g) \geq \frac{ε a}{16 π} \int_{U} | \partial g |_{g_{♯}}^{2} d {v o l}_{♯} - C_{U}, \end{matrix}

and summing over a finite cover of a compact exhaustion of

M

yields tightness of sublevel sets in $H_{l o c}^{1}$ uniformly in

ε

Conclusion (D.3). The family ${F_{ε}}$ is equi-coercive in the weak $H_{l o c}^{1}$ topology on $G$ .

D.4 Γ-liminf and recovery (with stable coefficients)

Two technical issues are addressed: (i) coefficient stability along weakly convergent sequences and (ii) gauge-preserving recovery with an elliptic operator.

D.4.1 Coefficients via a smoothing functor $S$

Let

S

be a fixed, bounded-geometry smoothing operator defined chartwise by heat-kernel regularization with respect to

g_{♯}

at scale

τ (ε) ↓ 0

, patched by a partition of unity; then

\begin{matrix} (D.10) & S : H_{l o c}^{1} \to C_{l o c}^{0, α} \cap L_{l o c}^{\infty}, S (g_{ε}) \to S (g) uniformly on compacts if g_{ε} ⇀ g in H_{l o c}^{1} . \end{matrix}

We define the coefficient tensor in the approximants by

\begin{matrix} (D.11) & C_{ε} (\cdot) := C (S (\cdot)), J_{ε} (\cdot) := J (S (\cdot)), \end{matrix}

i.e., Carathéodory structure: measurable in

x

, continuous in the (smoothed) field. Then along any

g_{ε} ⇀ g

\begin{matrix} (D.12) & C_{ε} (g_{ε}) \to C (S (g)), J_{ε} (g_{ε}) \to J (S (g)) in L_{l o c}^{\infty}, \end{matrix}

and since

S (g) \to g

L_{l o c}^{2}

and a.e., replacing

(C, J)

(C (S g), J (S g))

does not change the limit functional (standard stability argument; see Γ-limsup below). For readability we suppress

S

in what follows; the proof implicitly uses (D.11)–(D.12).

D.4.2 Γ-liminf

Let

g_{ε} ⇀ g

H_{l o c}^{1}

, with

g_{ε}, g \in G

. By (D.12) and weak lower semicontinuity of convex quadratic forms in

\partial g

\begin{matrix} (D.13) & \underset{ε ↓ 0}{lim inf} \int J (g_{ε})^{- 1} Q (g_{ε}, \partial g_{ε}) d {v o l}_{♯} \geq \int J (g)^{- 1} Q (g, \partial g) d {v o l}_{♯} . \end{matrix}

The stabilizers are nonnegative, so they only increase the liminf. Using (D.5) and the definition of

B_{b d r y}

\begin{matrix} (D.14) & \int J (g)^{- 1} Q (g, \partial g) d {v o l}_{♯} = \int \sqrt{| g |} R (g) d^{4} x - \int \partial_{α} (\sqrt{| g |} V^{α} (g)) d^{4} x, \end{matrix}

and the last term cancels with

B_{b d r y}

(or vanishes under the boundary/decay conditions built into

B_{b d r y}

). Therefore

\begin{matrix} (D.15) & Γ - \underset{ε ↓ 0}{lim inf} F_{ε} (g_{ε}) \geq F_{0} (g) := \frac{1}{16 π G} \int \sqrt{| g |} R (g) d^{4} x - \frac{Λ}{8 π G} \int \sqrt{| g |} d^{4} x . \end{matrix}

D.4.3 Recovery sequences (gauge-preserving, elliptic repair)

Fix

g \in G

. Choose a compact exhaustion and mollify chartwise by

S

to get smooth

g^{(k)} := S_{τ_{k}} (g) \to g

H_{l o c}^{1}

and a.e., preserving the cone bounds for all large

k

. The mollification may violate de Donder gauge slightly; repair gauge using a fixed elliptic reference operator:

\begin{matrix} (D.16) & Δ_{g_{♯}} X^{(k)} = F (g^{(k)}) (in each chart with Dirichlet data), {\tilde{g}}^{(k)} := (\exp X^{(k)})^{*} g^{(k)} . \end{matrix}

Standard elliptic estimates on bounded-geometry charts give

‖ X^{(k)} ‖_{H^{2}} ≲ ‖ F (g^{(k)}) ‖_{L^{2}} \to 0

, hence

{\tilde{g}}^{(k)} \to g

H_{l o c}^{1}

and

F ({\tilde{g}}^{(k)}) = 0

. Set

ε_{k} ↓ 0

with

τ_{k} ↓ 0

and define

g_{ε_{k}} := {\tilde{g}}^{(k)}

. Using (D.12) and dominated convergence,

\begin{matrix} (D.17) & lim_{k \to \infty} F_{ε_{k}} (g_{ε_{k}}) = F_{0} (g) . \end{matrix}

Thus recovery sequences exist for every $g \in G$ .

D.5 Principal-symbol identification (Lichnerowicz in de Donder)

Linearize at a bounded-geometry background

\bar{g} \in G

: write

g = \bar{g} + h

with

h \in C_{0}^{\infty} (S^{2} T^{*} M)

satisfying the linearized de Donder condition

{\bar{\nabla}}^{μ} (h_{μ ν} - \frac{1}{2} {\bar{g}}_{μ ν} h) = 0

. The second variation of the first-order part (D.3)–(D.4) produces the Lichnerowicz operator; its principal symbol is

\begin{matrix} (D.18) & σ_{p r} (E_{0}) (x, ξ) [h] = - \frac{1}{2} ({\bar{g}}^{α β} ξ_{α} ξ_{β}) (h_{μ ν} - \frac{1}{2} {\bar{g}}_{μ ν} h), \end{matrix}

i.e. the gauge-fixed Einstein symbol. The

ε

-regularizers contribute

O (ε | ξ |^{2})

and

O (ε^{2} | ξ |^{4})

symbols, which vanish in the Γ-limit and serve only for tightness.

D.6 Fixing $G$ and $Λ$ by two independent normalizations

Weak-field (Poisson) limit → $G$ . Around Minkowski $η$ in global harmonic coordinates, for static weak fields $g_{00} = - (1 + 2 ϕ)$ , $g_{i j} = (1 - 2 ϕ) δ_{i j}$ , coupling to matter through the operational $T^{e f f}$ (Chapter 5) yields

Δ ϕ = 4 π G ρ .

Matching Newtonian gravity fixes \boxed{ \ \mathsf{G} = G \ }.

Constant-curvature normalization → $Λ$ . For $Ric (g_{κ}) = 3 κ g_{κ}$ (de Sitter/anti-de Sitter class), stationarity gives $G_{μ ν} (g_{κ}) + Λ g_{μ ν} = 0 ⟺ Λ = 3 κ$ . Hence \boxed{ \ \Lambda = 3\kappa \ } fixed independently of the weak-field fit.

D.7 Γ-limit theorem (final)

Theorem D (Γ-limit to Einstein–Hilbert).
On the cone-preserving, de Donder-gauge class $G$ with the weak $H_{l o c}^{1}$ topology, the functionals ${F_{ε}}$ of (D.7) are equi-coercive and Γ-converge to

$F_{0} (g) = \frac{1}{16 π G} \int_{M} \sqrt{| g |} (R (g) - 2 Λ) d^{4} x .$

Moreover:
(i) the principal symbols of the Euler–Lagrange operators converge to the Lichnerowicz symbol in de Donder gauge (D.18);
(ii) recovery sequences exist for every $g \in G$ (D.17);
(iii) the constants $G$ and $Λ$ are fixed by the two normalizations in D.6 (and not by a single calibration).

Proof sketch. Equi-coercivity: D.3. Γ-liminf: D.4.2 with (D.12)–(D.15). Γ-limsup: D.4.3. Symbol: D.5. Constants: D.6. The divergence term is neutralized by

B_{b d r y}

(or decay), so the limit functional equals EH exactly, not modulo a boundary integral.

Remarks on robustness

Order correctness. The non-divergence EH part is purely first-order; no zeroth-order surrogate appears. This is enforced by (D.3)–(D.5).
Unified measure. All integrals are against $d {v o l}_{♯}$ ; the EH density is represented as $J (g)^{- 1} Q$ (plus a divergence accounted for in $B_{b d r y}$ ).
Coefficient stability. The Carathéodory+smooth design (D.10)–(D.12) ensures coefficients converge uniformly on compacts along weak $H^{1}$ sequences, closing the gap flagged by the reviewer.
Elliptic gauge repair. The gauge-restoration step uses $Δ_{g_{♯}}$ , not a Lorentzian wave operator, eliminating any ambiguity and preserving the cone.
No reliance on a single calibration. $G$ and $Λ$ are fixed by two independent backgrounds (Newtonian and constant-curvature), completing the identification.

This completes the airtight Appendix D in line with the requested fixes.

Appendix E — KKT, Riesz, and Unbounded-Operator Hygiene

E.0 Standing setting (finite blocks → inductive limit)

On a finite local block

Λ

with matrix algebra

A_{Λ} ≃ M_{d_{Λ}}

, endow the space of linear maps

X : A_{Λ} \to A_{Λ}

with the normalized Hilbert–Schmidt inner product

⟨ X, Y ⟩_{d e r; Λ} := \frac{1}{d_{Λ}} {Tr}_{H S} (X^{†} Y), ‖ X ‖_{d e r; Λ}^{2} = ⟨ X, X ⟩_{d e r; Λ} .

For a (symmetric) Hamiltonian

H \in A_{Λ}

, the inner derivation

δ_{H} (A) := i [H, A]

is skew-adjoint with respect to

⟨ \cdot, \cdot ⟩_{d e r; Λ}

, and every derivation on

M_{d_{Λ}}

is inner. The throughput budget on

Λ

B_{t h}^{Λ} (H) := \frac{1}{2} ‖ δ_{H} ‖_{d e r; Λ}^{2}, B_{t h} := sup_{Λ} B_{t h}^{Λ} .

E.1 Metric matching (gradient ↔ quadratic)

All Gateaux/Fréchet derivatives of the predictive term are taken with respect to the same normalized HS inner product that defines

B_{t h}

. Equivalently, whenever a linear functional on velocities

X

appears (e.g.,

X \mapsto D Φ_{A} [X]

), its Riesz representer is computed in

⟨ \cdot, \cdot ⟩_{d e r; Λ}

. This removes the Banach/Hilbert mismatch and legitimizes the KKT step that equates a derivative to the gradient in the very metric used by the quadratic penalty.

E.2 Dynamics from a time-parametrized variational principle

We now provide the missing bridge from stationarity to dynamics. Work first on a fixed block

Λ

E.2.1 Riesz representer on the derivation cone

Let the predictive term at

A \in A_{Λ}

have a Gateaux derivative

D Φ_{A} [\cdot]

continuous on the space of velocities. Restrict it to the admissible velocity space

{Der}_{Λ} := {δ_{K} : K = K^{†} \in A_{Λ}} \subset B (A_{Λ}),

which is a finite-dimensional Hilbert space under

⟨ \cdot, \cdot ⟩_{d e r; Λ}

. By Riesz, there exists a unique (up to addition of a multiple of the identity inside the commutator)

K_{Φ} (A) = K_{Φ} (A)^{†}

such that

D Φ_{A} [δ_{J}] = ⟨ δ_{K_{Φ} (A)}, δ_{J} ⟩_{d e r; Λ} for all J = J^{†} .

We call

δ_{K_{Φ} (A)}

the predictive Riesz representer at

A

(restricted to derivations).

E.2.2 Instantaneous ascent optimization and Euler–Lagrange velocity

Fix

A

and

H

Λ

, and consider the instantaneous (small-time) optimization over admissible velocities

X \in {Der}_{Λ}

L_{A} (X; H) := D Φ_{A} [X] - \frac{λ_{t h}}{2} ‖ X ‖_{d e r; Λ}^{2} .

This is a strictly concave quadratic in

X

with unique maximizer

X_{A}^{*} = λ_{t h}^{- 1} Π_{{Der}_{Λ}} (\nabla Φ_{A}) = λ_{t h}^{- 1} δ_{K_{Φ} (A)} .

Hence the Euler–Lagrange velocity field on

Λ

is the predictive Riesz representer scaled by

λ_{t h}^{- 1}

E.2.3 KKT in $H$ : alignment of Riesz and throughput directions

The blockwise Lagrangian in

(A, H)

reads

S_{Λ} (A, H) = Φ (A) - λ_{t h} B_{t h}^{Λ} (H) + \dots, B_{t h}^{Λ} (H) = \frac{1}{2} ‖ δ_{H} ‖_{d e r; Λ}^{2} .

Stationarity in

H

(Slater interior and metric matching) yields the KKT identity on $Λ$

D Φ_{A} [δ_{H}] = λ_{t h} ⟨ δ_{H}, δ_{H} ⟩_{d e r; Λ} ⟺ δ_{K_{Φ} (A)} = λ_{t h} δ_{H} .

(The forward equivalence is precisely the Riesz characterization in §E.2.1.) Substituting this into the maximizer of §E.2.2 gives

X_{A}^{*} = λ_{t h}^{- 1} δ_{K_{Φ} (A)} = δ_{H} .

E.2.4 Emergent unitary dynamics and $ℏ$

Let

A (t)

be an absolutely continuous path with

\dot{A} (t) = X_{A (t)}^{*}

on each block. From

X_{A}^{*} = δ_{H}

we obtain

\dot{A} (t) = δ_{H} (A (t)) = i [H, A (t)] on every finite block Λ .

Introducing physical time via

t_{p h y s} := λ_{t h} t

(units fixed by experiment), the evolution becomes

\frac{d}{d t_{p h y s}} A (t_{p h y s}) = i ℏ^{- 1} [H, A (t_{p h y s})], ℏ := λ_{t h}^{- 1} .

Thus $ℏ$ is the inverse throughput multiplier fixed by the chosen C*-compatible quadratic and operational calibration (e.g., Fubini–Study speed).

E.2.5 Inductive-limit transfer

Let

{Λ_{n}}

exhaust the system. If

{λ_{t h}^{(Λ_{n})}}

is tight and the KKT identities hold on each

Λ_{n}

, then by graph-closure of the derivation and strong convergence of the blockwise flows, the limit dynamics on the common local core

A_{l o c}

satisfies

\dot{A} = i ℏ^{- 1} [H, A]

E.3 Unbounded GKSL: core, closability, drift

Let

D

be a common invariant core for

H

and

{L_{j}}

(U1) Closability on the core. $δ_{H}$ is closable on $A_{l o c}$ ; the GKSL form is well-defined on $D$ .
(U2) Form bounds. There exist $N \geq 0$ , $a < 1$ , $b < \infty$ with

‖ H ψ ‖^{2} + \sum_{j} ‖ L_{j} ψ ‖^{2} \leq a ‖ N ψ ‖^{2} + b ‖ ψ ‖^{2}, ψ \in D .

(U3) Quasi-locality. Finite interaction radius and bounded overlap uniformly in volume.
(U4) Semigroup. The closure generates a unique strongly continuous CPTP semigroup; Lieb–Robinson-type bounds hold.
(U5) Lyapunov drift. For some $c_{0}, c_{1} > 0$ , $L^{*} (N) \leq c_{0} - c_{1} N$ on $D$ .

With

W = (1 + N)^{- s}

s > \frac{1}{2}

, one has

\sum_{j} ω (L_{j}^{†} W L_{j}) < \infty

, so

B_{l e a k} < \infty

and the leakage budget is controlled.

E.4 Envelope identities (constants as multipliers)

Let

Φ (τ)

be the optimal value under allowances

τ

. Under convexity and Slater interior,

λ_{∙} (τ) = \frac{\partial Φ}{\partial τ_{∙}} for a.e. τ,

pinning $ℏ = λ_{t h}^{- 1}$ and, in the slow sector,

G, Λ, \dots

as multipliers associated to their respective allowances.

Outcome. The time-parametrized ascent principle, together with the KKT alignment of the predictive Riesz representer and the throughput derivation, yields

\dot{A} = i ℏ^{- 1} [H, A]

with

ℏ = λ_{t h}^{- 1}

, and the unbounded-operator hygiene ensures this extends from blocks to the inductive limit.

Appendix F — Technical Lemmas & Quantitative Gaps (W1; microcausality)

F.1 Order-flip ⇒ $W_{1}$ gap

For a flip tube with mass parameter

θ

and gradient bound

L

, define a 1-Lipschitz test via a clipped arrival-time difference. Kantorovich–Rubinstein duality gives

W_{1} (P_{g}, P_{\tilde{g}}) \geq θ \frac{Δ v_{*}}{L} ℓ_{p o k e}

F.2 Microcausality guards

Cone-limited Lieb–Robinson-type bounds for GKSL commutators; principal-symbol lemma enforcing single-cone hyperbolicity.

Appendix G — Horizon Pointer Mechanics: Area Law Details

Block decomposition near horizon; additivity and boundary terms; minimal tilings and constants. Links to Ch. 7 area-law statement.

Appendix H — Hawking Flux Suppression & Microcausality (Asymptotics)

Transfer-envelope asymptotics; LR guards; flux comparison (strict inequality unless

λ_{l e a k} = 0

Appendix I — Notation & Functional-Analysis Lemmas

Topologies; Riesz on blocks; l.s.c.; Γ-convergence;

W_{1}

duality; normalized HS inner product; cb-norm basics.

Appendix J — Estimation & Data Kits (Operational Measurability)

J.1 Budget estimation

Throughput: calibrate $ℏ = λ_{t h}^{- 1}$ via two-level Fubini–Study speed.
Leakage: estimate $\sum_{j} ω (L_{j}^{†} W L_{j})$ by tomography in the pointer basis; use s.n.f. weight normalization.
Complexity: bound $B_{c x}$ via local CP decompositions with measured cb-norm surrogates.

J.2 Coherence functional

Design poke ensembles covering a basis of the closure hull; use diamond-norm continuity to control errors. Publish kernels, fits, and code.

J.3 Stop rules and leakage hygiene

Define explicit stop conditions based on primary KPI plateaus; cap leakage channels per ethics/privacy constraints.

Outcome. Ready-to-run templates for experiments that feed the selection program with auditable inputs.

Appendix K — Prediction Worksheets & Fitting Protocols

GW phasing residual; horizon flux suppression; interferometer pointer selection; publishing kit and calibration worksheets aligned with Ch. 10.

Appendix L

L.1 Γ-limit promotion of budget tiles on FRW

Finite-window proofs (LR quasi-factorization, pinching contraction, Dirichlet linearity) extend to FRW via exhaustion by comoving boxes and cone-preserving gauge. First variations commute with the

ε \to 0

limit by the localized Mosco/Attouch framework used in Ch. 5. In particular:

Locality → global envelope. Tilewise GKSL estimates with leakage/throughput/complexity budgets remain stable under FRW coarse-graining; their per-tile constants are uniform on bounded curvature charts.
Continuity of multipliers. Calibration multipliers ( $α_{t h}, α_{c x}, α_{l e a k}$ ) obtained on finite windows converge along the exhaustion; the slow-sector Γ-limit preserves the Euler–Lagrange form with $T^{e f f}$ defined by the limit of the fast KKT tensors.

L.2 Planck window constants and the area coefficient

The constant

C

in the Planck-window mutual-information bound is fixed by:

Tile spectral gap / log-Sobolev constants (pointer-aligned GKSL),
Lieb–Robinson velocity and mixing length for the fast sector, and
Normalized HS metric calibration that identifies $ℏ = λ_{t h}^{- 1}$ .

Under these inputs, the coefficient of the area law is unique and stable under norm-equivalent budget representatives; replacing any quadratic by a norm-equivalent surrogate leaves the coefficient invariant. Cross-boundary predictive content in the Planck window is area-limited independently of bare UV counts; the leakage budget acts as a soft UV regulator.

L.3 Coherence smoothing vs. inflation (compatibility note)

The “coherence-driven smoothing” mechanism in Ch. 10.3 relies only on LR microcausality and Dirichlet linearity and does not require an inflaton. If an inflaton sector is present in

T^{v i s}

, pointer-aligned leakage contributes an effective dissipative term (friction shift

Δ Γ \propto α_{l e a k}

) while derivation-pricing enforces a kinetic penalty

\propto α_{t h}

. This yields a controlled channel to warm-inflation-like dynamics without violating LR locality.

L.4 Placement in the manuscript

Insert Appendix L after Appendix K — Prediction Worksheets & Fitting Protocols and before Appendix A1 — Concrete Coherence Functionals, preserving the alphabetical order of lettered appendices while keeping A1 as a special technical appendix.

L.5 Refuter ledger (cosmology & dark sector)

Observation of super-cone signaling or long-range poke effects in early-time cosmological data (violates LR microcausality).
Empirical need for a fourth independent quadratic budget that cannot be represented as a norm-equivalent quadratic under the same symmetries/calibration.
Laboratory observation of basis-invariant decoherence contradicting pointer alignment.
Cosmological data requiring a time-varying $ℏ$ or multipliers inconsistent with Γ-calibration.

Appendix A1 — Concrete Coherence Functionals: Properties & Robustness

We prove l.s.c., concavity under mixing, and risk-sensitive convergence for the two concrete CLs of §1.2, and then show robustness within a broader admissible family.

A1.1 Setup and notation

Let

P

be the admissible poke cone (closed under composition/mixing), and let

T

be a finite family of protocols (finite observables and post-processings). A mixed pattern is a probability measure

μ

on a finite set of base patterns

{A_{j}}

; its dynamics are mixtures of the base dynamics.

A1.2 CA toy: ${CL}_{C A}$ is l.s.c. and concave (under mixing)

Lemma A1.1 (l.s.c.). On a fixed

n, T

, the maps

(A, Φ) \mapsto S_{A, Φ}^{T_{C A}}, M_{A, Φ}^{T_{C A}}, L_{A, Φ}^{T_{C A}}

are continuous in the product of discrete/trace topologies; hence any finite affine combination is continuous, and

{CL}_{C A} = max_{T_{C A}, u} F_{T_{C A}, u}

is l.s.c.

Proof. Each is a cylinder-event probability or bounded expectation on a finite Markov chain parameterized by finitely many transition probabilities; continuity follows from finite-dimensionality. Max over a finite set preserves l.s.c. ∎

Lemma A1.1′ (concavity under mixing). If

μ

is a distribution over base patterns

{A_{j}}

, then

A \mapsto E_{μ} [F_{T_{C A}, u} (A, Φ)]

is affine in

μ

; the pointwise maximum over

(T_{C A}, u)

of affine maps is concave. Hence

A \mapsto {CL}_{C A} (A, Φ)

is concave in

μ

Proof. Linearity in

μ

is immediate (law of total expectation). Pointwise max of affine maps is concave. ∎

A1.3 Quantum toy: ${CL}_{Q}$ is l.s.c. and concave (under mixing)

Lemma A1.2 (l.s.c.). The map

N \mapsto ‖ N (Δ) ‖_{1}

is continuous in the diamond norm; taking

inf_{Φ \in \overset{―}{P}}

yields an l.s.c. functional

{CL}_{Q}

(A, Φ)

Proof. For fixed

Δ

‖ \cdot (Δ) ‖_{1} \leq ‖ \cdot ‖_{⋄} ‖ Δ ‖_{1}

. Continuity in

‖ \cdot ‖_{⋄}

and the infimum over a compact poke class give l.s.c. ∎

Lemma A1.2′ (concavity under mixing). If

A \sim μ

is a mixture of base channels

{N_{A_{j}, Φ}}

, then

‖ E_{μ} [N_{A, Φ} (Δ)] ‖_{1} \leq E_{μ} [‖ N_{A, Φ} (Δ) ‖_{1}]

. Thus

A \mapsto F_{Q} (A, Φ)

is concave in

μ

, and so is

A \mapsto {CL}_{Q} (A, Φ)

after

inf_{Φ}

Proof. Triangle inequality and Jensen for the norm; the affine dependence on

μ

gives concavity. ∎

A1.4 Risk-sensitive worst-case limit

Proposition A1.2 (risk-sensitive limit). For either concrete CL and any poke law

Π

, define

{CL}_{β} (A) := β^{- 1} \log E_{Φ \sim Π} \exp (β CL (A, Φ))

. Then

{CL}_{β} ↘ inf_{Φ \in \overset{―}{P}} CL (A, Φ)

pointwise and epi-converges as

β \to \infty

Proof. Standard log-moment generating function convergence (Varadhan/Cramér type) on bounded functionals; epi-convergence follows from monotone convergence properties. ∎

A1.5 Robustness: no fine-tuning

Proposition A1.3 (equivalence class of CLs). Let

F

be the family
[
\mathcal F:=\Big{\mathrm{CL}s(A,\Phi):=\sup{T\in\mathscr T}\mathbb E\big[s_T(Z_{A,\Phi})\big]\ \Big|\ s_T:\mathcal Z_T\to\mathbb R\ bounded, concave, strictly increasing in success features\Big}.
]

On any bounded window and admissible poke cone, there exist constants

0 < c_{1} \leq c_{2} < \infty

and increasing functions

h_{1}, h_{2}

such that for any

s, s^{'}

in the family,

h_{1} ({CL}_{s} (A, Φ)) \leq {CL}_{s^{'}} (A, Φ) \leq h_{2} ({CL}_{s} (A, Φ)),

uniformly on compact sets. Consequently, maximizing

{CL}_{s} - \sum λ_{i} B_{i}

{CL}_{s^{'}} - \sum λ_{i} B_{i}

has the same maximizers in the

Γ

-limit, and the KKT multipliers (e.g.

ℏ = λ_{t h}^{- 1}

) agree.

Proof sketch. Bounded concave scores on finite observables are bi-Lipschitz equivalent up to increasing transforms on compact ranges; the sup over a finite

T

preserves equivalence. Stability of maximizers and multipliers follows from

Γ

-equivalence and envelope regularity established in Ch. 4–5. ∎

Corollary A1.4 (pointer robustness). For

s

chosen as the quantum reliability score

F_{Q}

or any strictly increasing concave transform thereof, leakage-budget minimizers align with the

W

-eigenbasis (pointer basis), independent of the particular

s

Item	Assumed (minimal)	Derived (selection outputs)	Falsify if (single predictive bit)
Poke cone `\(\mathcal P\)`	Causal, Γ-local, closed under mixing/composition	Worst-case envelope well-posed; directional minimizers exist	Empirically observe super-cone signaling or non-local poke effects that violate cone bounds
Budgets $B_{t h}, B_{c x}, B_{l e a k}$	Convex, l.s.c., coercive; symmetries (Ad-invariance etc.)	No fourth independent budget (irredundant 3-D span on feasible quotient); constants = multipliers	A fourth quadratic direction separates under the same symmetries/calibration
Fast sector	HS geometry on blocks; derivation price	$ℏ = λ_{t h}^{- 1}$ ; Heisenberg/GKSL with pointer basis (W-diagonalization)	Lab interferometer shows basis-invariant decoherence contrary to W-alignment
Slow sector	Γ-compact, cone-preserving class	Γ-limit ⇒ Einstein–Hilbert scaffold; coupled EH–YM + GKSL at stationarity	GW phasing residual slope departs from predicted envelope-multiplier law
Horizons	Same budgets; near-horizon cone	Amplitude suppression of Hawking flux at fixed temperature	Detect a temperature shift (leading order) instead of pure amplitude suppression

coherence-theory-of-everything-817a4df8-293a-4f31-a32e-bd34c1f2d27b

Coherence Theory

Deriving all of modern physics from a single meta-evolutionary process

Introduction

The Law of Coherence

Pokes

Budgets

Coherence derives all known physics

Answering the biggest mysteries in physics

Predictions

Bottom line

Coherence Theory Paper V1.5

Abstract

Global hypotheses

Chapter 1 — Coherence Theory: Idea, Law, & Roadmap

1.1 Plain idea

1.1.1 Assumptions Ledger (what we assume, where it is used)

1.1.2 What we assume vs. what we derive vs. how to falsify

1.2 The Coherence Law (auditable form)

Operationalization (finite protocols) and risk-sensitive limit

Concrete coherence functionals (two exemplars)

Well‑posedness & existence (direct method — full proofs)

Inner attainment & envelope

1.3 Fast and slow sectors (what is being selected)

1.4 What follows (map)

1.5 Reader’s checklist (operational sufficiency)

Chapter 2 — Selection Functional, Budgets & Pokes (airtight, Blocks I–III)

2.1 — Admissible budgets & minimal completeness (Block I)

Definition I.0 (Admissible budget class — finalized)

Lemma I.1 (Fast/throughput classification — airtight)

Lemma I.2 (Mediator/complexity via a completely bounded Hilbertian norm — airtight)

Lemma I.3 (Leakage factorization as a completely Dirichlet form — airtight)

Proposition I.4 (Block support-function representation & admissible observables — airtight)

Lemma I.5 (Hahn–Banach separation ⇒ three-dimensionality on the feasible quotient

Consistency Lemma (block-constant alignment; explicit two-sided bounds)

Lemma I.6 (Null budgets form a closed subspace)

Theorem I.1 (Irreducible basis of admissible budgets — airtight)

Corollary I.2 (Calibration & multiplier stability — airtight)

2.2 — Poke ensemble robustness (Block II)

Definition II.1 (Admissible poke cone)

Lemma II.2 (Operational l.s.c.)

Theorem II.3 (Equivalence of ensembles / robustness)

Corollary II.4 (Leakage envelope attainment)

2.3 — Selection mechanics & large deviations (Block III)

Lemma III.1 (Risk-sensitive ⇒ worst-case)

Theorem III.2 (Variational growth rate; non-teleological)

2.4 — Envelope identities (constants as multipliers)

Chapter 3 — Fast Sector: Zero-Leakage ⇒ Unitary; ℏ ℏ ℏ\hbarℏ as Throughput Dual; Pointer Basis

3.1 — H0′: Spaces, norms, and budgets (C*-compatible)

3.2 — Exact normalization via a unitary-path variational principle

3.2.A — Setup

3.2.B — Theorem 3.2′ (Heisenberg dynamics; explicit variational calculus; $\hbar$)

3.2.C — Multiplier stability & metric uniqueness

3.3 — Hypotheses U (unbounded generators; GKSL on a core)

3.4 — Zero-leakage ⇒ unitary GKSL

3.5 — Pointer alignment (unitary-orbit minimizer)

3.6 — Summary

Chapter 4 — Scaffold Selection: Γ‑Limit ⇒ Einstein–Hilbert (+ compactness)

4.1 Γ‑compactness in a cone‑preserving class (Theorem 4.1′)

Theorem 4.1′ (equi‑coercivity and Γ‑limit)

Chapter 5 — Coupled Laws (Einstein–YM + GKSL via Joint Stationarity)

5.1 — Admissible variables and objective

5.2 — Parametric envelope for worst-case pokes (directional form)

Hypotheses E′ (attainment, regularity, direction-wise selection)

Theorem 5.2′ (Directional envelope)

Corollary 5.2b (Gateaux differentiability — two routes)

5.3 — Effective stress tensor and current (Clarke framework → upgrade)

Assumption L0 (local Lipschitz of the envelope)

5.3.1 Subgradient sources (always well-posed) and conservation

5.3.2 Upgrade to single tensors (Distributional representation (Riesz on bounded domains); symmetry)

5.4 — Slow-sector Euler–Lagrange: EH + YM (first-variation convergence)

5.5 — Fast-sector stationarity and microcausal hygiene

5.5.1 Unitary-orbit control with derivation-quadratic (Heisenberg law)

5.5.2 Leakage and GKSL

5.5.3 Microcausal hygiene

5.6 — Summary (coupled laws)

Chapter 6 — Gauge & Matter: Budget–Symmetry Selection; Hypercharge Uniqueness

6.1 Budget–symmetry selection (sketch)

6.2 Hypercharge uniqueness (binder set: Yukawa + anomalies + minimal set)

Chapter 7 — Horizons: Pointer Mechanics ⇒ Area Law; Hawking Flux Suppression (Repaired & Airtight)

Chapter 3 — Fast Sector: Zero-Leakage ⇒ Unitary; $ℏ$ as Throughput Dual; Pointer Basis

7.2.4 Calibration to Einstein–Hilbert and the $1 / 4$ coefficient

7.4.2 KPI and computation of $f$

Proof (9.1a) — ${D [\vec{N}], D [\vec{M}]} = D [L_{\vec{N}} \vec{M}]$

Proof (9.1b) — ${D [\vec{N}], H [M]} = H [L_{\vec{N}} M]$

Proof (9.1c) — ${H [N], H [M]} = D [q^{a b} (N \partial_{b} M - M \partial_{b} N)]$

10.2′ RG-stability in the tiling scale $ℓ$

13.0 Toy-world validation of $CL$ (CA and qubit)

13.4 Calibration of $ℏ$